Daily Papers

During developmental processes such as embryogenesis, how a group of cells fold into specific structures, is a central question in biology that defines how living organisms form. Establishing tissue-level morphology critically relies on how every single cell decides to position itself relative to its neighboring cells. Despite its importance, it remains a major challenge to understand and predict the behavior of every cell within the living tissue over time during such intricate processes. To tackle this question, we propose a geometric deep learning model that can predict multicellular folding and embryogenesis, accurately capturing the highly convoluted spatial interactions among cells. We demonstrate that multicellular data can be represented with both granular and foam-like physical pictures through a unified graph data structure, considering both cellular interactions and cell junction networks. We successfully use our model to achieve two important tasks, interpretable 4-D morphological sequence alignment, and predicting local cell rearrangements before they occur at single-cell resolution. Furthermore, using an activation map and ablation studies, we demonstrate that cell geometries and cell junction networks together regulate local cell rearrangement which is critical for embryo morphogenesis. This approach provides a novel paradigm to study morphogenesis, highlighting a unified data structure and harnessing the power of geometric deep learning to accurately model the mechanisms and behaviors of cells during development. It offers a pathway toward creating a unified dynamic morphological atlas for a variety of developmental processes such as embryogenesis.

Large language models (LLMs) demonstrate strong performance, but they often lack transparency. We introduce GeoLAN, a training framework that treats token representations as geometric trajectories and applies stickiness conditions inspired by recent developments related to the Kakeya Conjecture. We have developed two differentiable regularizers, Katz-Tao Convex Wolff (KT-CW) and Katz-Tao Attention (KT-Attn), that promote isotropy and encourage diverse attention. Our experiments with Gemma-3 (1B, 4B, 12B) and Llama-3-8B show that GeoLAN frequently maintains task accuracy while improving geometric metrics and reducing certain fairness biases. These benefits are most significant in mid-sized models. Our findings reveal scale-dependent trade-offs between geometric precision and performance, suggesting that geometry-aware training is a promising approach to enhance mechanistic interpretability.

The CLEVR dataset has been used extensively in language grounded visual reasoning in Machine Learning (ML) and Natural Language Processing (NLP) domains. We present a graph parser library for CLEVR, that provides functionalities for object-centric attributes and relationships extraction, and construction of structural graph representations for dual modalities. Structural order-invariant representations enable geometric learning and can aid in downstream tasks like language grounding to vision, robotics, compositionality, interpretability, and computational grammar construction. We provide three extensible main components - parser, embedder, and visualizer that can be tailored to suit specific learning setups. We also provide out-of-the-box functionality for seamless integration with popular deep graph neural network (GNN) libraries. Additionally, we discuss downstream usage and applications of the library, and how it accelerates research for the NLP research community.

Massive molecular simulations of drug-target proteins have been used as a tool to understand disease mechanism and develop therapeutics. This work focuses on learning a generative neural network on a structural ensemble of a drug-target protein, e.g. SARS-CoV-2 Spike protein, obtained from computationally expensive molecular simulations. Model tasks involve characterizing the distinct structural fluctuations of the protein bound to various drug molecules, as well as efficient generation of protein conformations that can serve as an complement of a molecular simulation engine. Specifically, we present a geometric autoencoder framework to learn separate latent space encodings of the intrinsic and extrinsic geometries of the protein structure. For this purpose, the proposed Protein Geometric AutoEncoder (ProGAE) model is trained on the protein contact map and the orientation of the backbone bonds of the protein. Using ProGAE latent embeddings, we reconstruct and generate the conformational ensemble of a protein at or near the experimental resolution, while gaining better interpretability and controllability in term of protein structure generation from the learned latent space. Additionally, ProGAE models are transferable to a different state of the same protein or to a new protein of different size, where only the dense layer decoding from the latent representation needs to be retrained. Results show that our geometric learning-based method enjoys both accuracy and efficiency for generating complex structural variations, charting the path toward scalable and improved approaches for analyzing and enhancing high-cost simulations of drug-target proteins.

Geometric foundation models show promise in 3D reconstruction, yet their progress is severely constrained by the scarcity of diverse, large-scale 3D annotations. While Internet videos offer virtually unlimited raw data, utilizing them as a scaling source for geometric learning is challenging due to the absence of ground-truth geometry and the presence of observational noise. To address this, we propose SAGE, a framework for Scalable Adaptation of GEometric foundation models from raw video streams. SAGE leverages a hierarchical mining pipeline to transform videos into training trajectories and hybrid supervision: (1) Informative training trajectory selection; (2) Sparse Geometric Anchoring via SfM point clouds for global structural guidance; and (3) Dense Differentiable Consistency via 3D Gaussian rendering for multi-view constraints. To prevent catastrophic forgetting, we introduce a regularization strategy using anchor data. Extensive experiments show that SAGE significantly enhances zero-shot generalization, reducing Chamfer Distance by 20-42% on unseen benchmarks (7Scenes, TUM-RGBD, Matterport3D) compared to state-of-the-art baselines. To our knowledge, SAGE pioneers the adaptation of geometric foundation models via Internet video, establishing a scalable paradigm for general-purpose 3D learning.

Audio-visual zero-shot learning aims to classify samples consisting of a pair of corresponding audio and video sequences from classes that are not present during training. An analysis of the audio-visual data reveals a large degree of hyperbolicity, indicating the potential benefit of using a hyperbolic transformation to achieve curvature-aware geometric learning, with the aim of exploring more complex hierarchical data structures for this task. The proposed approach employs a novel loss function that incorporates cross-modality alignment between video and audio features in the hyperbolic space. Additionally, we explore the use of multiple adaptive curvatures for hyperbolic projections. The experimental results on this very challenging task demonstrate that our proposed hyperbolic approach for zero-shot learning outperforms the SOTA method on three datasets: VGGSound-GZSL, UCF-GZSL, and ActivityNet-GZSL achieving a harmonic mean (HM) improvement of around 3.0%, 7.0%, and 5.3%, respectively.

Despite being the workhorse of deep learning, the backpropagation algorithm is no panacea. It enforces sequential layer updates, thus preventing efficient parallelization of the training process. Furthermore, its biological plausibility is being challenged. Alternative schemes have been devised; yet, under the constraint of synaptic asymmetry, none have scaled to modern deep learning tasks and architectures. Here, we challenge this perspective, and study the applicability of Direct Feedback Alignment to neural view synthesis, recommender systems, geometric learning, and natural language processing. In contrast with previous studies limited to computer vision tasks, our findings show that it successfully trains a large range of state-of-the-art deep learning architectures, with performance close to fine-tuned backpropagation. At variance with common beliefs, our work supports that challenging tasks can be tackled in the absence of weight transport.

Many geometric learning problems require invariants on heterogeneous product spaces, i.e., products of distinct spaces carrying different group actions, where standard techniques do not directly apply. We show that, when a group G acts transitively on a space M, any G-invariant function on a product space X times M can be reduced to an invariant of the isotropy subgroup H of M acting on X alone. Our approach establishes an explicit orbit equivalence (X times M)/G cong X/H, yielding a principled reduction that preserves expressivity. We apply this characterization to Equivariant Neural Fields, extending them to arbitrary group actions and homogeneous conditioning spaces, and thereby removing the major structural constraints imposed by existing methods.

Safe autonomous navigation requires a holistic understanding of dynamic environments, necessitating the simultaneous estimation of metric depth, semantic segmentation, and instance trajectories. While depth-aware video panoptic segmentation (DVPS) unifies these tasks, existing approaches often rely on computationally expensive, multi-stage pipelines or offline tracking, rendering them unsuitable for real-time decision-making. To address this, we propose DVPSFormer, a unified online architecture designed for efficient 4D scene understanding. Central to our approach is explicit scene discretization (ESD), a novel mechanism that leverages segmentation queries to represent foreground and background regions, enabling a discrete-to-continuous (D2C) depth head to decode metric depth in a single pass. This tightly couples semantic and geometric learning while significantly reducing latency. Furthermore, we propose an online majority voting (OMV) mechanism that exploits temporal consistency to refine classification during instance tracking. DVPSFormer establishes a new state-of-the-art on the Cityscapes-DVPS and SemKITTI-DVPS benchmarks, offering a streamlined solution for online robotic perception. Code and models are available at https://royyang0714.github.io/DVPSFormer.

Processing information on 3D objects requires methods stable to rigid-body transformations, in particular rotations, of the input data. In image processing tasks, convolutional neural networks achieve this property using rotation-equivariant operations. However, contrary to images, graphs generally have irregular topology. This makes it challenging to define a rotation-equivariant convolution operation on these structures. In this work, we propose Spherical Graph Convolutional Network (S-GCN) that processes 3D models of proteins represented as molecular graphs. In a protein molecule, individual amino acids have common topological elements. This allows us to unambiguously associate each amino acid with a local coordinate system and construct rotation-equivariant spherical filters that operate on angular information between graph nodes. Within the framework of the protein model quality assessment problem, we demonstrate that the proposed spherical convolution method significantly improves the quality of model assessment compared to the standard message-passing approach. It is also comparable to state-of-the-art methods, as we demonstrate on Critical Assessment of Structure Prediction (CASP) benchmarks. The proposed technique operates only on geometric features of protein 3D models. This makes it universal and applicable to any other geometric-learning task where the graph structure allows constructing local coordinate systems.

Efficiently synthesizing novel views from sparse inputs while maintaining accuracy remains a critical challenge in 3D reconstruction. While advanced techniques like radiance fields and 3D Gaussian Splatting achieve rendering quality and impressive efficiency with dense view inputs, they suffer from significant geometric reconstruction errors when applied to sparse input views. Moreover, although recent methods leverage monocular depth estimation to enhance geometric learning, their dependence on single-view estimated depth often leads to view inconsistency issues across different viewpoints. Consequently, this reliance on absolute depth can introduce inaccuracies in geometric information, ultimately compromising the quality of scene reconstruction with Gaussian splats. In this paper, we present RDG-GS, a novel sparse-view 3D rendering framework with Relative Depth Guidance based on 3D Gaussian Splatting. The core innovation lies in utilizing relative depth guidance to refine the Gaussian field, steering it towards view-consistent spatial geometric representations, thereby enabling the reconstruction of accurate geometric structures and capturing intricate textures. First, we devise refined depth priors to rectify the coarse estimated depth and insert global and fine-grained scene information to regular Gaussians. Building on this, to address spatial geometric inaccuracies from absolute depth, we propose relative depth guidance by optimizing the similarity between spatially correlated patches of depth and images. Additionally, we also directly deal with the sparse areas challenging to converge by the adaptive sampling for quick densification. Across extensive experiments on Mip-NeRF360, LLFF, DTU, and Blender, RDG-GS demonstrates state-of-the-art rendering quality and efficiency, making a significant advancement for real-world application.

We propose a spherical kernel for efficient graph convolution of 3D point clouds. Our metric-based kernels systematically quantize the local 3D space to identify distinctive geometric relationships in the data. Similar to the regular grid CNN kernels, the spherical kernel maintains translation-invariance and asymmetry properties, where the former guarantees weight sharing among similar local structures in the data and the latter facilitates fine geometric learning. The proposed kernel is applied to graph neural networks without edge-dependent filter generation, making it computationally attractive for large point clouds. In our graph networks, each vertex is associated with a single point location and edges connect the neighborhood points within a defined range. The graph gets coarsened in the network with farthest point sampling. Analogous to the standard CNNs, we define pooling and unpooling operations for our network. We demonstrate the effectiveness of the proposed spherical kernel with graph neural networks for point cloud classification and semantic segmentation using ModelNet, ShapeNet, RueMonge2014, ScanNet and S3DIS datasets. The source code and the trained models can be downloaded from https://github.com/hlei-ziyan/SPH3D-GCN.

Neural representations of 3D data have been widely adopted across various applications, particularly in recent work leveraging coordinate-based networks to model scalar or vector fields. However, these approaches face inherent challenges, such as handling thin structures and non-watertight geometries, which limit their flexibility and accuracy. In contrast, we propose a novel geometric data representation that models geometry as distributions-a powerful representation that makes no assumptions about surface genus, connectivity, or boundary conditions. Our approach uses diffusion models with a novel network architecture to learn surface point distributions, capturing fine-grained geometric details. We evaluate our representation qualitatively and quantitatively across various object types, demonstrating its effectiveness in achieving high geometric fidelity. Additionally, we explore applications using our representation, such as textured mesh representation, neural surface compression, dynamic object modeling, and rendering, highlighting its potential to advance 3D geometric learning.

The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation. While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications. Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.

In document image rectification, there exist rich geometric constraints between the distorted image and the ground truth one. However, such geometric constraints are largely ignored in existing advanced solutions, which limits the rectification performance. To this end, we present DocGeoNet for document image rectification by introducing explicit geometric representation. Technically, two typical attributes of the document image are involved in the proposed geometric representation learning, i.e., 3D shape and textlines. Our motivation arises from the insight that 3D shape provides global unwarping cues for rectifying a distorted document image while overlooking the local structure. On the other hand, textlines complementarily provide explicit geometric constraints for local patterns. The learned geometric representation effectively bridges the distorted image and the ground truth one. Extensive experiments show the effectiveness of our framework and demonstrate the superiority of our DocGeoNet over state-of-the-art methods on both the DocUNet Benchmark dataset and our proposed DIR300 test set. The code is available at https://github.com/fh2019ustc/DocGeoNet.

Geometric deep learning, i.e., designing neural networks to handle the ubiquitous geometric data such as point clouds and graphs, have achieved great successes in the last decade. One critical inductive bias is that the model can maintain invariance towards various transformations such as translation, rotation, and scaling. The existing graph neural network (GNN) approaches can only maintain permutation-invariance, failing to guarantee invariance with respect to other transformations. Besides GNNs, other works design sophisticated transformation-invariant layers, which are computationally expensive and difficult to be extended. To solve this problem, we revisit why the existing neural networks cannot maintain transformation invariance when handling geometric data. Our findings show that transformation-invariant and distance-preserving initial representations are sufficient to achieve transformation invariance rather than needing sophisticated neural layer designs. Motivated by these findings, we propose Transformation Invariant Neural Networks (TinvNN), a straightforward and general framework for geometric data. Specifically, we realize transformation-invariant and distance-preserving initial point representations by modifying multi-dimensional scaling before feeding the representations into neural networks. We prove that TinvNN can strictly guarantee transformation invariance, being general and flexible enough to be combined with the existing neural networks. Extensive experimental results on point cloud analysis and combinatorial optimization demonstrate the effectiveness and general applicability of our proposed method. Based on the experimental results, we advocate that TinvNN should be considered a new starting point and an essential baseline for further studies of transformation-invariant geometric deep learning.

Brain-computer interfaces (BCIs) offer transformative potential, but decoding neural signals presents significant challenges. The core premise of this paper is built around demonstrating methods to elucidate the underlying low-dimensional geometric structure present in high-dimensional brainwave data in order to assist in downstream BCI-related neural classification tasks. We demonstrate two pipelines related to electroencephalography (EEG) signal processing: (1) a preliminary pipeline removing noise from individual EEG channels, and (2) a downstream manifold learning pipeline uncovering geometric structure across networks of EEG channels. We conduct preliminary validation using two EEG datasets and situate our demonstration in the context of the BCI-relevant imagined digit decoding problem. Our preliminary pipeline uses an attention-based EEG filtration network to extract clean signal from individual EEG channels. Our primary pipeline uses a fast Fourier transform, a Laplacian eigenmap, a discrete analog of Ricci flow via Ollivier's notion of Ricci curvature, and a graph convolutional network to perform dimensionality reduction on high-dimensional multi-channel EEG data in order to enable regularizable downstream classification. Our system achieves competitive performance with existing signal processing and classification benchmarks; we demonstrate a mean test correlation coefficient of >0.95 at 2 dB on semi-synthetic neural denoising and a downstream EEG-based classification accuracy of 0.97 on distinguishing digit- versus non-digit- thoughts. Results are preliminary and our geometric machine learning pipeline should be validated by more extensive follow-up studies; generalizing these results to larger inter-subject sample sizes, different hardware systems, and broader use cases will be crucial.

We consider the hierarchical representation of documents as graphs and use geometric deep learning to classify them into different categories. While graph neural networks can efficiently handle the variable structure of hierarchical documents using the permutation invariant message passing operations, we show that we can gain extra performance improvements using our proposed selective graph pooling operation that arises from the fact that some parts of the hierarchy are invariable across different documents. We applied our model to classify clinical trial (CT) protocols into completed and terminated categories. We use bag-of-words based, as well as pre-trained transformer-based embeddings to featurize the graph nodes, achieving f1-scores around 0.85 on a publicly available large scale CT registry of around 360K protocols. We further demonstrate how the selective pooling can add insights into the CT termination status prediction. We make the source code and dataset splits accessible.

In this paper, we introduce a novel way to use geometric deep learning for knot data by constructing a functor that takes knots to graphs and using graph neural networks. We will attempt to predict several knot invariants with this approach. This approach demonstrates high generalization capabilities.

Multimodal Large Language Models (MLLMs) excel at 2D semantic understanding but lack intrinsic 3D awareness, resulting in representations that fail to maintain geometric and spatial consistency across video frames. Given the scarcity of large-scale 3D data, we present GeoVR, a novel framework that learns geometric representations using purely 2D video sequences. This approach effectively restructures the semantic latent space within MLLMs to unlock spatial intelligence. Rather than employing superficial feature mixing, GeoVR reshapes the internal representations of the MLLM by distilling geometry knowledge from pre-trained 3D foundation models. This is accomplished through a multi-objective learning strategy driven by four complementary geometric targets: (1) estimating inter-frame camera poses to embed varying viewpoint dynamics, (2) regressing dense depth maps to anchor physical distances, (3) predicting a metric scale factor for real-world calibration, and (4) distilling multi-scale 3D features to align the intermediate feature space. Guided by these explicit physical and geometric constraints, the model's internal representations naturally develop strong 3D awareness. Extensive experiments on spatial reasoning benchmarks demonstrate that GeoVR achieves state-of-the-art performance, establishing a new paradigm for endowing foundation models with spatial intelligence.

Video spatial reasoning is essential for navigation-oriented perception and long-video question answering, where models must infer spatial relations across long horizons under changing viewpoints. However, existing multimodal large language models (MLLMs) remain largely semantic-centric, and often fail to reliably aggregate consistent spatial evidence from redundant video observations, leading to inefficient or unstable reasoning. To address these issues, we propose ConsiSpace, a geometry-consistency-aware framework for geometry-sensitive video spatial reasoning that turns spatial consistency into both an evidence organization principle and an explicit post-SFT learning signal. We build a geometry-consistent memory (GCM) including implicit evidence tokens and explicit geometric cues, and leverage efficient organization strategies to compactly preserve task-related spatial evidence. Furthermore, we utilize unified consistency self-supervised reinforcement learning (UC-SSRL) after supervised fine-tuning to improve cross-view stability, with answer-, metric-, and topology-consistency rewards. Extensive experiments on three spatial-reasoning benchmarks, VSI-Bench, OSI-Bench, and MMSI-Video-Bench, show consistent gains, improving the average score by 12.6 points over the strongest baselines.

Predicting how a drug-like molecule binds to a specific protein target is a core problem in drug discovery. An extremely fast computational binding method would enable key applications such as fast virtual screening or drug engineering. Existing methods are computationally expensive as they rely on heavy candidate sampling coupled with scoring, ranking, and fine-tuning steps. We challenge this paradigm with EquiBind, an SE(3)-equivariant geometric deep learning model performing direct-shot prediction of both i) the receptor binding location (blind docking) and ii) the ligand's bound pose and orientation. EquiBind achieves significant speed-ups and better quality compared to traditional and recent baselines. Further, we show extra improvements when coupling it with existing fine-tuning techniques at the cost of increased running time. Finally, we propose a novel and fast fine-tuning model that adjusts torsion angles of a ligand's rotatable bonds based on closed-form global minima of the von Mises angular distance to a given input atomic point cloud, avoiding previous expensive differential evolution strategies for energy minimization.

The inference of topological principles is a key problem in structured reconstruction. We observe that wrongly predicted topological relationships are often incurred by the lack of holistic geometry clues in low-level features. Inspired by the fact that massive signals can be compactly described with frequency analysis, we experimentally explore the efficiency and tendency of learning structure geometry in the frequency domain. Accordingly, we propose a frequency-domain feature learning strategy (F-Learn) to fuse scattered geometric fragments holistically for topology-intact structure reasoning. Benefiting from the parsimonious design, the F-Learn strategy can be easily deployed into a deep reconstructor with a lightweight model modification. Experiments demonstrate that the F-Learn strategy can effectively introduce structure awareness into geometric primitive detection and topology inference, bringing significant performance improvement to final structured reconstruction. Code and pre-trained models are available at https://github.com/Geo-Tell/F-Learn.

Computational RNA design tasks are often posed as inverse problems, where sequences are designed based on adopting a single desired secondary structure without considering 3D geometry and conformational diversity. We introduce gRNAde, a geometric RNA design pipeline operating on 3D RNA backbones to design sequences that explicitly account for structure and dynamics. Under the hood, gRNAde is a multi-state Graph Neural Network that generates candidate RNA sequences conditioned on one or more 3D backbone structures where the identities of the bases are unknown. On a single-state fixed backbone re-design benchmark of 14 RNA structures from the PDB identified by Das et al. [2010], gRNAde obtains higher native sequence recovery rates (56% on average) compared to Rosetta (45% on average), taking under a second to produce designs compared to the reported hours for Rosetta. We further demonstrate the utility of gRNAde on a new benchmark of multi-state design for structurally flexible RNAs, as well as zero-shot ranking of mutational fitness landscapes in a retrospective analysis of a recent ribozyme. Open source code: https://github.com/chaitjo/geometric-rna-design

Large Language Models frequently hallucinate in precision-critical domains such as technical diagramming and mechanical design, where outputs must satisfy strict geometric constraints. We study open-ended geometric synthesis from natural language: translating free-form descriptions into precise constructions whose entities must simultaneously satisfy dozens of interacting constraints. To make this tractable, we release PyGeoX, a programmable geometric DSL that compiles declarative constraints into a differentiable loss, and PyGeoX-Bench, a stratified suite of 300 problems with per-constraint verifiable rewards. Using PyGeoX as a verifier, we identify a failure mode we call Outlier Gradient Masking: under global-norm rewards (any scheme that aggregates residuals through a single norm, for example, exp(-MSE)), a single outlier constraint can nullify the learning signal across all others. To address this, we propose Saturating Additive Rewards (SAR), which decompose the reward into bounded per-constraint terms, preserving partial progress and ensuring consistent gradients even under severe violations. Against MSE-based rewards, the natural baseline for geometry solvers, SAR improves the hard-tier solving rate by 2.3times, and the resulting 8B model is competitive with much larger frontier systems on this benchmark. We release the engine, benchmark, and data at https://github.com/Huawei-AI4Math/PyGeoX.

Recent advances in neural reconstruction using posed image sequences have made remarkable progress. However, due to the lack of depth information, existing volumetric-based techniques simply duplicate 2D image features of the object surface along the entire camera ray. We contend this duplication introduces noise in empty and occluded spaces, posing challenges for producing high-quality 3D geometry. Drawing inspiration from traditional multi-view stereo methods, we propose an end-to-end 3D neural reconstruction framework CVRecon, designed to exploit the rich geometric embedding in the cost volumes to facilitate 3D geometric feature learning. Furthermore, we present Ray-contextual Compensated Cost Volume (RCCV), a novel 3D geometric feature representation that encodes view-dependent information with improved integrity and robustness. Through comprehensive experiments, we demonstrate that our approach significantly improves the reconstruction quality in various metrics and recovers clear fine details of the 3D geometries. Our extensive ablation studies provide insights into the development of effective 3D geometric feature learning schemes. Project page: https://cvrecon.ziyue.cool/

Reverse engineering and rapid prototyping of computer-aided design (CAD) models from 3D scans, sketches, or simple text prompts are vital in industrial product design. However, recent advances in geometric deep learning techniques lack a multi-modal understanding of parametric CAD features stored in their boundary representation (BRep). This study presents the largest compilation of 10 million multi-modal annotations and metadata for 1 million ABC CAD models, namely A2Z, to unlock an unprecedented level of BRep learning. A2Z comprises (i) high-resolution meshes with salient 3D scanning features, (ii) 3D hand-drawn sketches equipped with (iii) geometric and topological information about BRep co-edges, corners, and surfaces, and (iv) textual captions and tags describing the product in the mechanical world. Creating such carefully structured, large-scale data, which requires nearly 5 terabytes of storage to leverage unparalleled CAD learning/retrieval tasks, is very challenging. The scale, quality, and diversity of our multi-modal annotations are assessed using novel metrics, GPT-5, Gemini, and extensive human feedback mechanisms. To this end, we also merge an additional 25,000 CAD models of electronic enclosures (e.g., tablets, ports) designed by skilled professionals with our A2Z dataset. Subsequently, we train and benchmark a foundation model on a subset of 150K CAD models to detect BRep co-edges and corner vertices from 3D scans, a key downstream task in CAD reverse engineering. The annotated dataset, metrics, and checkpoints will be publicly released to support numerous research directions.

FourCastNet 3 advances global weather modeling by implementing a scalable, geometric machine learning (ML) approach to probabilistic ensemble forecasting. The approach is designed to respect spherical geometry and to accurately model the spatially correlated probabilistic nature of the problem, resulting in stable spectra and realistic dynamics across multiple scales. FourCastNet 3 delivers forecasting accuracy that surpasses leading conventional ensemble models and rivals the best diffusion-based methods, while producing forecasts 8 to 60 times faster than these approaches. In contrast to other ML approaches, FourCastNet 3 demonstrates excellent probabilistic calibration and retains realistic spectra, even at extended lead times of up to 60 days. All of these advances are realized using a purely convolutional neural network architecture tailored for spherical geometry. Scalable and efficient large-scale training on 1024 GPUs and more is enabled by a novel training paradigm for combined model- and data-parallelism, inspired by domain decomposition methods in classical numerical models. Additionally, FourCastNet 3 enables rapid inference on a single GPU, producing a 60-day global forecast at 0.25{\deg}, 6-hourly resolution in under 4 minutes. Its computational efficiency, medium-range probabilistic skill, spectral fidelity, and rollout stability at subseasonal timescales make it a strong candidate for improving meteorological forecasting and early warning systems through large ensemble predictions.

Cosmological simulations provide a wealth of data in the form of point clouds and directed trees. A crucial goal is to extract insights from this data that shed light on the nature and composition of the Universe. In this paper we introduce CosmoBench, a benchmark dataset curated from state-of-the-art cosmological simulations whose runs required more than 41 million core-hours and generated over two petabytes of data. CosmoBench is the largest dataset of its kind: it contains 34 thousand point clouds from simulations of dark matter halos and galaxies at three different length scales, as well as 25 thousand directed trees that record the formation history of halos on two different time scales. The data in CosmoBench can be used for multiple tasks -- to predict cosmological parameters from point clouds and merger trees, to predict the velocities of individual halos and galaxies from their collective positions, and to reconstruct merger trees on finer time scales from those on coarser time scales. We provide several baselines on these tasks, some based on established approaches from cosmological modeling and others rooted in machine learning. For the latter, we study different approaches -- from simple linear models that are minimally constrained by symmetries to much larger and more computationally-demanding models in deep learning, such as graph neural networks. We find that least-squares fits with a handful of invariant features sometimes outperform deep architectures with many more parameters and far longer training times. Still there remains tremendous potential to improve these baselines by combining machine learning and cosmology to fully exploit the data. CosmoBench sets the stage for bridging cosmology and geometric deep learning at scale. We invite the community to push the frontier of scientific discovery by engaging with this dataset, available at https://cosmobench.streamlit.app

Much of the success of deep learning is drawn from building architectures that properly respect underlying symmetry and structure in the data on which they operate - a set of considerations that have been united under the banner of geometric deep learning. Often problems in the physical sciences deal with relatively small sets of points in two- or three-dimensional space wherein translation, rotation, and permutation equivariance are important or even vital for models to be useful in practice. In this work, we present rotation- and permutation-equivariant architectures for deep learning on these small point clouds, composed of a set of products of terms from the geometric algebra and reductions over those products using an attention mechanism. The geometric algebra provides valuable mathematical structure by which to combine vector, scalar, and other types of geometric inputs in a systematic way to account for rotation invariance or covariance, while attention yields a powerful way to impose permutation equivariance. We demonstrate the usefulness of these architectures by training models to solve sample problems relevant to physics, chemistry, and biology.

Energy-based predictive world models provide a powerful approach for multi-step visual planning by reasoning over latent energy landscapes rather than generating pixels. However, existing approaches face two major challenges: (i) their latent representations are typically learned in Euclidean space, neglecting the underlying geometric and hierarchical structure among states, and (ii) they struggle with long-horizon prediction, which leads to rapid degradation across extended rollouts. To address these challenges, we introduce GeoWorld, a geometric world model that preserves geometric structure and hierarchical relations through a Hyperbolic JEPA, which maps latent representations from Euclidean space onto hyperbolic manifolds. We further introduce Geometric Reinforcement Learning for energy-based optimization, enabling stable multi-step planning in hyperbolic latent space. Extensive experiments on CrossTask and COIN demonstrate around 3% SR improvement in 3-step planning and 2% SR improvement in 4-step planning compared to the state-of-the-art V-JEPA 2. Project website: https://steve-zeyu-zhang.github.io/GeoWorld.

Transformers serve as the backbone architectures of Foundational Models, where a domain-specific tokenizer helps them adapt to various domains. Graph Transformers (GTs) have recently emerged as a leading model in geometric deep learning, outperforming Graph Neural Networks (GNNs) in various graph learning tasks. However, the development of tokenizers for graphs has lagged behind other modalities, with existing approaches relying on heuristics or GNNs co-trained with Transformers. To address this, we introduce GQT (Graph Quantized Tokenizer), which decouples tokenizer training from Transformer training by leveraging multi-task graph self-supervised learning, yielding robust and generalizable graph tokens. Furthermore, the GQT utilizes Residual Vector Quantization (RVQ) to learn hierarchical discrete tokens, resulting in significantly reduced memory requirements and improved generalization capabilities. By combining the GQT with token modulation, a Transformer encoder achieves state-of-the-art performance on 16 out of 18 benchmarks, including large-scale homophilic and heterophilic datasets. The code is available at: https://github.com/limei0307/graph-tokenizer

We present our position on the elusive quest for a general-purpose framework for specifying and studying deep learning architectures. Our opinion is that the key attempts made so far lack a coherent bridge between specifying constraints which models must satisfy and specifying their implementations. Focusing on building a such a bridge, we propose to apply category theory -- precisely, the universal algebra of monads valued in a 2-category of parametric maps -- as a single theory elegantly subsuming both of these flavours of neural network design. To defend our position, we show how this theory recovers constraints induced by geometric deep learning, as well as implementations of many architectures drawn from the diverse landscape of neural networks, such as RNNs. We also illustrate how the theory naturally encodes many standard constructs in computer science and automata theory.

Despite the impressive progress on understanding and generating images shown by the recent unified architectures, the integration of 3D tasks remains challenging and largely unexplored. In this paper, we introduce UniUGG, the first unified understanding and generation framework for 3D modalities. Our unified framework employs an LLM to comprehend and decode sentences and 3D representations. At its core, we propose a spatial decoder leveraging a latent diffusion model to generate high-quality 3D representations. This allows for the generation and imagination of 3D scenes based on a reference image and an arbitrary view transformation, while remaining supports for spatial visual question answering (VQA) tasks. Additionally, we propose a geometric-semantic learning strategy to pretrain the vision encoder. This design jointly captures the input's semantic and geometric cues, enhancing both spatial understanding and generation. Extensive experimental results demonstrate the superiority of our method in visual representation, spatial understanding, and 3D generation. The source code will be released upon paper acceptance.

Geometric Deep Learning has recently attracted significant interest in a wide range of machine learning fields, including document analysis. The application of Graph Neural Networks (GNNs) has become crucial in various document-related tasks since they can unravel important structural patterns, fundamental in key information extraction processes. Previous works in the literature propose task-driven models and do not take into account the full power of graphs. We propose Doc2Graph, a task-agnostic document understanding framework based on a GNN model, to solve different tasks given different types of documents. We evaluated our approach on two challenging datasets for key information extraction in form understanding, invoice layout analysis and table detection. Our code is freely accessible on https://github.com/andreagemelli/doc2graph.

Geometric Deep Learning has recently made striking progress with the advent of continuous Deep Implicit Fields. They allow for detailed modeling of watertight surfaces of arbitrary topology while not relying on a 3D Euclidean grid, resulting in a learnable parameterization that is not limited in resolution. Unfortunately, these methods are often not suitable for applications that require an explicit mesh-based surface representation because converting an implicit field to such a representation relies on the Marching Cubes algorithm, which cannot be differentiated with respect to the underlying implicit field. In this work, we remove this limitation and introduce a differentiable way to produce explicit surface mesh representations from Deep Signed Distance Functions. Our key insight is that by reasoning on how implicit field perturbations impact local surface geometry, one can ultimately differentiate the 3D location of surface samples with respect to the underlying deep implicit field. We exploit this to define MeshSDF, an end-to-end differentiable mesh representation which can vary its topology. We use two different applications to validate our theoretical insight: Single-View Reconstruction via Differentiable Rendering and Physically-Driven Shape Optimization. In both cases our differentiable parameterization gives us an edge over state-of-the-art algorithms.

Geometric Deep Learning has recently made striking progress with the advent of continuous deep implicit fields. They allow for detailed modeling of watertight surfaces of arbitrary topology while not relying on a 3D Euclidean grid, resulting in a learnable parameterization that is unlimited in resolution. Unfortunately, these methods are often unsuitable for applications that require an explicit mesh-based surface representation because converting an implicit field to such a representation relies on the Marching Cubes algorithm, which cannot be differentiated with respect to the underlying implicit field. In this work, we remove this limitation and introduce a differentiable way to produce explicit surface mesh representations from Deep Implicit Fields. Our key insight is that by reasoning on how implicit field perturbations impact local surface geometry, one can ultimately differentiate the 3D location of surface samples with respect to the underlying deep implicit field. We exploit this to define DeepMesh - an end-to-end differentiable mesh representation that can vary its topology. We validate our theoretical insight through several applications: Single view 3D Reconstruction via Differentiable Rendering, Physically-Driven Shape Optimization, Full Scene 3D Reconstruction from Scans and End-to-End Training. In all cases our end-to-end differentiable parameterization gives us an edge over state-of-the-art algorithms.

We present STream3R, a novel approach to 3D reconstruction that reformulates pointmap prediction as a decoder-only Transformer problem. Existing state-of-the-art methods for multi-view reconstruction either depend on expensive global optimization or rely on simplistic memory mechanisms that scale poorly with sequence length. In contrast, STream3R introduces an streaming framework that processes image sequences efficiently using causal attention, inspired by advances in modern language modeling. By learning geometric priors from large-scale 3D datasets, STream3R generalizes well to diverse and challenging scenarios, including dynamic scenes where traditional methods often fail. Extensive experiments show that our method consistently outperforms prior work across both static and dynamic scene benchmarks. Moreover, STream3R is inherently compatible with LLM-style training infrastructure, enabling efficient large-scale pretraining and fine-tuning for various downstream 3D tasks. Our results underscore the potential of causal Transformer models for online 3D perception, paving the way for real-time 3D understanding in streaming environments. More details can be found in our project page: https://nirvanalan.github.io/projects/stream3r.

Cancer is the second leading cause of death, with chemotherapy as one of the primary forms of treatment. As a result, researchers are turning to drug combination therapy to decrease drug resistance and increase efficacy. Current methods of drug combination screening, such as in vivo and in vitro, are inefficient due to stark time and monetary costs. In silico methods have become increasingly important for screening drugs, but current methods are inaccurate and generalize poorly to unseen anticancer drugs. In this paper, I employ a geometric deep-learning model utilizing a graph attention network that is equivariant to 3D rotations, translations, and reflections with structural motifs. Additionally, the gene expression of cancer cell lines is utilized to classify synergistic drug combinations specific to each cell line. I compared the proposed geometric deep learning framework to current state-of-the-art (SOTA) methods, and the proposed model architecture achieved greater performance on all 12 benchmark tasks performed on the DrugComb dataset. Specifically, the proposed framework outperformed other SOTA methods by an accuracy difference greater than 28%. Based on these results, I believe that the equivariant graph attention network's capability of learning geometric data accounts for the large performance improvements. The model's ability to generalize to foreign drugs is thought to be due to the structural motifs providing a better representation of the molecule. Overall, I believe that the proposed equivariant geometric deep learning framework serves as an effective tool for virtually screening anticancer drug combinations for further validation in a wet lab environment. The code for this work is made available online at: https://github.com/WeToTheMoon/EGAT_DrugSynergy.

Accurate document layout analysis remains a critical bottleneck for document parsing systems, due to the intricate coupling among heterogeneous document layout elements, geometric distortions (\eg, paper warping and bending, perspective variations), and reading order within diverse layout structures. Existing approaches typically rely on fragmented multi-stage pipelines or computationally heavy generative Transformer architectures, leading to error propagation and limited efficiency. In this paper, we present RT-DocLayout, a highly efficient end-to-end framework for document layout analysis, designed as a front-end for document parsing tasks. The proposed model unifies classification, detection, pixel-level segmentation, and reading order prediction for layout elements within a single 33M-parameter architecture. Built upon the RT-DETR, our key contribution is a unified multi-task formulation within a single query-based decoder that simultaneously classifies, regresses bounding box, generates masks, and constructs relationship to reason reading order. By jointly learning geometric and structural representations, RT-DocLayout introduces multi-task optimization that substantially improves robustness under real-world document distortions. Extensive experiments on public benchmarks demonstrate state-of-the-art performance in document layout analysis while maintaining real-time inference speed(132.1 FPS). When coupled with downstream OCR engines, RT-DocLayout significantly improves full-document reconstruction quality, providing a scalable and practical foundation for real-world document intelligence systems.

Molecular docking is a key computational tool utilized to predict the binding conformations of small molecules to protein targets, which is fundamental in the design of novel drugs. Despite recent advancements in geometric deep learning-based approaches leading to improvements in blind docking efficiency, these methods have encountered notable challenges, such as limited generalization performance on unseen proteins, the inability to concurrently address the settings of blind docking and site-specific docking, and the frequent occurrence of physical implausibilities such as inter-molecular steric clash. In this study, we introduce DeltaDock, a robust and versatile framework designed for efficient molecular docking to overcome these challenges. DeltaDock operates in a two-step process: rapid initial complex structures sampling followed by multi-scale iterative refinement of the initial structures. In the initial stage, to sample accurate structures with high efficiency, we develop a ligand-dependent binding site prediction model founded on large protein models and graph neural networks. This model is then paired with GPU-accelerated sampling algorithms. The sampled structures are updated using a multi-scale iterative refinement module that captures both protein-ligand atom-atom interactions and residue-atom interactions in the following stage. Distinct from previous geometric deep learning methods that are conditioned on the blind docking setting, DeltaDock demonstrates superior performance in both blind docking and site-specific docking settings. Comprehensive experimental results reveal that DeltaDock consistently surpasses baseline methods in terms of docking accuracy. Furthermore, it displays remarkable generalization capabilities and proficiency for predicting physically valid structures, thereby attesting to its robustness and reliability in various scenarios.

Recent works in geometric deep learning have introduced neural networks that allow performing inference tasks on three-dimensional geometric data by defining convolution, and sometimes pooling, operations on triangle meshes. These methods, however, either consider the input mesh as a graph, and do not exploit specific geometric properties of meshes for feature aggregation and downsampling, or are specialized for meshes, but rely on a rigid definition of convolution that does not properly capture the local topology of the mesh. We propose a method that combines the advantages of both types of approaches, while addressing their limitations: we extend a primal-dual framework drawn from the graph-neural-network literature to triangle meshes, and define convolutions on two types of graphs constructed from an input mesh. Our method takes features for both edges and faces of a 3D mesh as input and dynamically aggregates them using an attention mechanism. At the same time, we introduce a pooling operation with a precise geometric interpretation, that allows handling variations in the mesh connectivity by clustering mesh faces in a task-driven fashion. We provide theoretical insights of our approach using tools from the mesh-simplification literature. In addition, we validate experimentally our method in the tasks of shape classification and shape segmentation, where we obtain comparable or superior performance to the state of the art.

The Laplace-Beltrami operator (LBO) emerges from studying manifolds equipped with a Riemannian metric. It is often called the Swiss army knife of geometry processing as it allows to capture intrinsic shape information and gives rise to heat diffusion, geodesic distances, and a multitude of shape descriptors. It also plays a central role in geometric deep learning. In this work, we explore Finsler manifolds as a generalization of Riemannian manifolds. We revisit the Finsler heat equation and derive a Finsler heat kernel and a Finsler-Laplace-Beltrami Operator (FLBO): a novel theoretically justified anisotropic Laplace-Beltrami operator (ALBO). In experimental evaluations we demonstrate that the proposed FLBO is a valuable alternative to the traditional Riemannian-based LBO and ALBOs for spatial filtering and shape correspondence estimation. We hope that the proposed Finsler heat kernel and the FLBO will inspire further exploration of Finsler geometry in the computer vision community.

Design of neural networks that incorporate symmetry is crucial for geometric deep learning. Central to this effort is the development of invariant and equivariant operations. This works presents a systematic method for constructing valid invariant and equivariant operations. It can handle inputs and outputs in the form of Cartesian tensors with different rank, as well as spherical tensors with different types. In addition, our method features a graphical representation utilizing the symmetric tensor network, which simplifies both the proofs and constructions related to invariant and equivariant functions. We also apply this approach to design the equivariant interaction message for the geometry graph neural network, and equivariant machine learning model to learn the constitutive law of materials.

Learning to predict agent motions with relationship reasoning is important for many applications. In motion prediction tasks, maintaining motion equivariance under Euclidean geometric transformations and invariance of agent interaction is a critical and fundamental principle. However, such equivariance and invariance properties are overlooked by most existing methods. To fill this gap, we propose EqMotion, an efficient equivariant motion prediction model with invariant interaction reasoning. To achieve motion equivariance, we propose an equivariant geometric feature learning module to learn a Euclidean transformable feature through dedicated designs of equivariant operations. To reason agent's interactions, we propose an invariant interaction reasoning module to achieve a more stable interaction modeling. To further promote more comprehensive motion features, we propose an invariant pattern feature learning module to learn an invariant pattern feature, which cooperates with the equivariant geometric feature to enhance network expressiveness. We conduct experiments for the proposed model on four distinct scenarios: particle dynamics, molecule dynamics, human skeleton motion prediction and pedestrian trajectory prediction. Experimental results show that our method is not only generally applicable, but also achieves state-of-the-art prediction performances on all the four tasks, improving by 24.0/30.1/8.6/9.2%. Code is available at https://github.com/MediaBrain-SJTU/EqMotion.

Photorealistic human novel view synthesis from a single image is crucial for democratizing immersive 3D telepresence, eliminating the need for complex multi-camera setups. However, current rendering-centric methods prioritize visual fidelity over explicit geometric understanding and struggle with intricate regions like faces and hands, leading to temporal instability. Meanwhile, human-centric frameworks suffer from memory bottlenecks since they typically rely on an auxiliary model to provide informative structural priors for geometric modeling, which limits real-time performance. To address these challenges, we propose PrismMirror, a geometry-guided framework for instant frontal view synthesis from a single image. By avoiding external geometric modeling and focusing on frontal view synthesis, our model optimizes visual integrity for telepresence. Specifically, PrismMirror introduces a novel cascade learning strategy that enables coarse-to-fine geometric feature learning. It first directly learns coarse geometric features, such as SMPL-X meshes and point clouds, and then refines textures through rendering supervision. To achieve real-time efficiency, we distill this unified framework into a lightweight linear attention model. Notably, PrismMirror is the first monocular human frontal view synthesis model that achieves real-time inference at 24 FPS, significantly outperforming previous methods in both visual authenticity and structural accuracy.

3D reconstruction from satellite imagery is essential for large-scale topographic analysis, yet the lack of high-fidelity training datasets with accurate occlusion labels remains a primary bottleneck. Existing benchmarks, such as US3D and WHU-Stereo, face inherent challenges in spatio-temporal mismatch -- environmental changes and shadow displacements between multi-view acquisitions -- and provide ambiguous ground truth in occluded regions due to LiDAR sparsity. In this paper, we propose SatUnreal, a high-precision synthetic dataset designed to fundamentally overcome these limitations through an Unreal Engine-based simulation pipeline. SatUnreal provides 10,000 stereo pairs with high resolution (0.3m GSD) and is characterized by: (1) Physical Geometry Simulation, replicating realistic satellite orbits by systematically varying baselines and azimuths; (2) Spatio-temporal Consistency, eliminating temporal noise through fixed virtual environments; (3) Topographic Diversity, spanning dense urban canyons to low-texture natural terrains; and (4) Mathematical Label Integrity, utilizing a novel two-step linetrace algorithm to generate flawless occlusion masks. Experimental results using SOTA iterative models demonstrate that models trained exclusively on SatUnreal achieve superior zero-shot transfer performance on real-world benchmarks (US3D, WHU-Stereo) compared to those trained on real datasets. Our findings prove that physically accurate synthetic data provides a more effective supervisory signal for learning geometric features than complex real-world observations, establishing a new paradigm for Sim-to-Real transfer in Earth Observation. Code and dataset are available at https://github.com/jmp-Telepix/SatUnreal_A_High-Precision_Synthetic_Dataset_for_Satellite_Stereo_Matching_via_UnrealEngine

Identifying key patterns of tactics implemented by rival teams, and developing effective responses, lies at the heart of modern football. However, doing so algorithmically remains an open research challenge. To address this unmet need, we propose TacticAI, an AI football tactics assistant developed and evaluated in close collaboration with domain experts from Liverpool FC. We focus on analysing corner kicks, as they offer coaches the most direct opportunities for interventions and improvements. TacticAI incorporates both a predictive and a generative component, allowing the coaches to effectively sample and explore alternative player setups for each corner kick routine and to select those with the highest predicted likelihood of success. We validate TacticAI on a number of relevant benchmark tasks: predicting receivers and shot attempts and recommending player position adjustments. The utility of TacticAI is validated by a qualitative study conducted with football domain experts at Liverpool FC. We show that TacticAI's model suggestions are not only indistinguishable from real tactics, but also favoured over existing tactics 90% of the time, and that TacticAI offers an effective corner kick retrieval system. TacticAI achieves these results despite the limited availability of gold-standard data, achieving data efficiency through geometric deep learning.

Facial landmark detection is a fundamental problem in computer vision for many downstream applications. This paper introduces a new facial landmark detector based on vision transformers, which consists of two unique designs: Dual Vision Transformer (D-ViT) and Long Skip Connections (LSC). Based on the observation that the channel dimension of feature maps essentially represents the linear bases of the heatmap space, we propose learning the interconnections between these linear bases to model the inherent geometric relations among landmarks via Channel-split ViT. We integrate such channel-split ViT into the standard vision transformer (i.e., spatial-split ViT), forming our Dual Vision Transformer to constitute the prediction blocks. We also suggest using long skip connections to deliver low-level image features to all prediction blocks, thereby preventing useful information from being discarded by intermediate supervision. Extensive experiments are conducted to evaluate the performance of our proposal on the widely used benchmarks, i.e., WFLW, COFW, and 300W, demonstrating that our model outperforms the previous SOTAs across all three benchmarks.

Quantum Implicit Neural Representations (QINRs) include components for learning and execution on gate-based quantum computers. While QINRs recently emerged as a promising new paradigm, many challenges concerning their architecture and ansatz design, the utility of quantum-mechanical properties, training efficiency and the interplay with classical modules remain. This paper advances the field by introducing a new type of QINR for 2D image and 3D geometric field learning, which we collectively refer to as Quantum Visual Field (QVF). QVF encodes classical data into quantum statevectors using neural amplitude encoding grounded in a learnable energy manifold, ensuring meaningful Hilbert space embeddings. Our ansatz follows a fully entangled design of learnable parametrised quantum circuits, with quantum (unitary) operations performed in the real Hilbert space, resulting in numerically stable training with fast convergence. QVF does not rely on classical post-processing -- in contrast to the previous QINR learning approach -- and directly employs projective measurement to extract learned signals encoded in the ansatz. Experiments on a quantum hardware simulator demonstrate that QVF outperforms the existing quantum approach and widely used classical foundational baselines in terms of visual representation accuracy across various metrics and model characteristics, such as learning of high-frequency details. We also show applications of QVF in 2D and 3D field completion and 3D shape interpolation, highlighting its practical potential.

Modern AI technologies for drug discovery are distributed across heterogeneous platforms-including web applications, desktop environments, and code libraries-leading to fragmented workflows, inconsistent interfaces, and high integration overhead. We present an agentic end-to-end drug design framework that leverages a Large Language Model (LLM) in conjunction with the Model Context Protocol (MCP) to dynamically coordinate access to biochemical databases, modular toolchains, and task-specific AI models. The system integrates four state-of-the-art components: MaSIF (MaSIF-site and MaSIF-seed-search) for geometric deep learning-based identification of protein-protein interaction (PPI) sites, Rosetta for grafting protein fragments onto protein backbones to form mini proteins, ProteinMPNN for amino acid sequences redesign, and AlphaFold3 for near-experimental accuracy in complex structure prediction. Starting from a target structure, the framework supports de novo binder generation via surface analysis, scaffold grafting and pose construction, sequence optimization, and structure prediction. Additionally, by replacing rigid, script-based workflows with a protocol-driven, LLM-coordinated architecture, the framework improves reproducibility, reduces manual overhead, and ensures extensibility, portability, and auditability across the entire drug design process.

Crash simulation is a cornerstone of modern vehicle development because it reduces the need for costly physical prototypes, accelerates safety-driven design iteration, and increasingly supports virtual testing workflows. At the same time, modeling structural crash mechanics remains exceptionally challenging: the response is governed by nonlinear contact, large deformation, material plasticity, failure, and complex multi-body interactions evolving over space and time on high-resolution finite-element meshes. In this work, we introduce CarCrashNet, a public high-fidelity open-source benchmark for data-driven structural crash simulation. CarCrashNet combines component-scale and full-vehicle simulations in a multi-modal format, including more than 14{,}000 bumper-beam pole-impact simulations with varying geometry, materials, and boundary conditions, together with 825 full-vehicle crash simulations built from three industry-standard vehicle models of increasing structural complexity: Dodge Neon, Toyota Yaris, and Chevrolet Silverado. To establish the reliability of the benchmark, we validate our open-source finite-element workflow based on OpenRadioss against both experimental crash data and the commercial solver Ansys LS-DYNA. We also introduce CrashSolver, a machine-learning model designed for full-vehicle crash prediction from high-resolution finite-element crash data. We further perform extensive benchmarking across the released datasets and evaluate CrashSolver against state-of-the-art geometric deep learning and transformer-based neural solvers. Our results position CarCrashNet as a foundation for reproducible research in structural simulation, crashworthiness modeling, and AI-driven virtual crash testing. The dataset is available at https://github.com/Mohamedelrefaie/CarCrashNet.

Conformal symmetries, i.e.\ coordinate transformations that preserve angles, play a key role in many fields, including physics, mathematics, computer vision and (geometric) machine learning. Here we build a neural network that is equivariant under general conformal transformations. To achieve this, we lift data from flat Euclidean space to Anti de Sitter (AdS) space. This allows us to exploit a known correspondence between conformal transformations of flat space and isometric transformations on the AdS space. We then build upon the fact that such isometric transformations have been extensively studied on general geometries in the geometric deep learning literature. We employ message-passing layers conditioned on the proper distance, yielding a computationally efficient framework. We validate our model on tasks from computer vision and statistical physics, demonstrating strong performance, improved generalization capacities, and the ability to extract conformal data such as scaling dimensions from the trained network.

Structure-based drug design (SBDD) leverages the 3D structure of biomolecular targets to guide the creation of new therapeutic agents. Recent advances in generative models, including diffusion models and geometric deep learning, have demonstrated promise in optimizing ligand generation. However, the scarcity of high-quality protein-ligand complex data and the inherent challenges in aligning generated ligands with target proteins limit the effectiveness of these methods. We propose BoKDiff, a novel framework that enhances ligand generation by combining multi-objective optimization and Best-of-K alignment methodologies. Built upon the DecompDiff model, BoKDiff generates diverse candidates and ranks them using a weighted evaluation of molecular properties such as QED, SA, and docking scores. To address alignment challenges, we introduce a method that relocates the center of mass of generated ligands to their docking poses, enabling accurate sub-component extraction. Additionally, we integrate a Best-of-N (BoN) sampling approach, which selects the optimal ligand from multiple generated candidates without requiring fine-tuning. BoN achieves exceptional results, with QED values exceeding 0.6, SA scores above 0.75, and a success rate surpassing 35%, demonstrating its efficiency and practicality. BoKDiff achieves state-of-the-art results on the CrossDocked2020 dataset, including a -8.58 average Vina docking score and a 26% success rate in molecule generation. This study is the first to apply Best-of-K alignment and Best-of-N sampling to SBDD, highlighting their potential to bridge generative modeling with practical drug discovery requirements. The code is provided at https://github.com/khodabandeh-ali/BoKDiff.git.

Interpretability methods are valuable only if their explanations faithfully describe the explained model. In this work, we consider neural networks whose predictions are invariant under a specific symmetry group. This includes popular architectures, ranging from convolutional to graph neural networks. Any explanation that faithfully explains this type of model needs to be in agreement with this invariance property. We formalize this intuition through the notion of explanation invariance and equivariance by leveraging the formalism from geometric deep learning. Through this rigorous formalism, we derive (1) two metrics to measure the robustness of any interpretability method with respect to the model symmetry group; (2) theoretical robustness guarantees for some popular interpretability methods and (3) a systematic approach to increase the invariance of any interpretability method with respect to a symmetry group. By empirically measuring our metrics for explanations of models associated with various modalities and symmetry groups, we derive a set of 5 guidelines to allow users and developers of interpretability methods to produce robust explanations.

Due to high utility in many applications, from social networks to blockchain to power grids, deep learning on non-Euclidean objects such as graphs and manifolds, coined Geometric Deep Learning (GDL), continues to gain an ever increasing interest. We propose a new L\'evy Flights Graph Convolutional Networks (LFGCN) method for semi-supervised learning, which casts the L\'evy Flights into random walks on graphs and, as a result, allows both to accurately account for the intrinsic graph topology and to substantially improve classification performance, especially for heterogeneous graphs. Furthermore, we propose a new preferential P-DropEdge method based on the Girvan-Newman argument. That is, in contrast to uniform removing of edges as in DropEdge, following the Girvan-Newman algorithm, we detect network periphery structures using information on edge betweenness and then remove edges according to their betweenness centrality. Our experimental results on semi-supervised node classification tasks demonstrate that the LFGCN coupled with P-DropEdge accelerates the training task, increases stability and further improves predictive accuracy of learned graph topology structure. Finally, in our case studies we bring the machinery of LFGCN and other deep networks tools to analysis of power grid networks - the area where the utility of GDL remains untapped.

In this paper, we present DV-Matcher, a novel learning-based framework for estimating dense correspondences between non-rigidly deformable point clouds. Learning directly from unstructured point clouds without meshing or manual labelling, our framework delivers high-quality dense correspondences, which is of significant practical utility in point cloud processing. Our key contributions are two-fold: First, we propose a scheme to inject prior knowledge from pre-trained vision models into geometric feature learning, which effectively complements the local nature of geometric features with global and semantic information; Second, we propose a novel deformation-based module to promote the extrinsic alignment induced by the learned correspondences, which effectively enhances the feature learning. Experimental results show that our method achieves state-of-the-art results in matching non-rigid point clouds in both near-isometric and heterogeneous shape collection as well as more realistic partial and noisy data.

Automated personality and soft skill assessment from multimodal behavioral data remains challenging due to limited datasets and methods that fail to capture geometric structure inherent in human traits. We introduce RecruitView, a dataset of 2,011 naturalistic video interview clips from 300+ participants with 27,000 pairwise comparative judgments across 12 dimensions: Big Five personality traits, overall personality score, and six interview performance metrics. To leverage this data, we propose Cross-Modal Regression with Manifold Fusion (CRMF), a geometric deep learning framework that explicitly models behavioral representations across hyperbolic, spherical, and Euclidean manifolds. CRMF employs geometry-specific expert networks to capture hierarchical trait structures, directional behavioral patterns, and continuous performance variations simultaneously. An adaptive routing mechanism dynamically weights expert contributions based on input characteristics. Through principled tangent space fusion, CRMF achieves superior performance while training 40-50% fewer trainable parameters than large multimodal models. Extensive experiments demonstrate that CRMF substantially outperforms the selected baselines, achieving up to 11.4% improvement in Spearman correlation and 6.0% in concordance index. Our RecruitView dataset is publicly available at https://huggingface.co/datasets/AI4A-lab/RecruitView

We introduce SldprtNet, a large-scale dataset comprising over 242,000 industrial parts, designed for semantic-driven CAD modeling, geometric deep learning, and the training and fine-tuning of multimodal models for 3D design. The dataset provides 3D models in both .step and .sldprt formats to support diverse training and testing. To enable parametric modeling and facilitate dataset scalability, we developed supporting tools, an encoder and a decoder, which support 13 types of CAD commands and enable lossless transformation between 3D models and a structured text representation. Additionally, each sample is paired with a composite image created by merging seven rendered views from different viewpoints of the 3D model, effectively reducing input token length and accelerating inference. By combining this image with the parameterized text output from the encoder, we employ the lightweight multimodal language model Qwen2.5-VL-7B to generate a natural language description of each part's appearance and functionality. To ensure accuracy, we manually verified and aligned the generated descriptions, rendered images, and 3D models. These descriptions, along with the parameterized modeling scripts, rendered images, and 3D model files, are fully aligned to construct SldprtNet. To assess its effectiveness, we fine-tuned baseline models on a dataset subset, comparing image-plus-text inputs with text-only inputs. Results confirm the necessity and value of multimodal datasets for CAD generation. It features carefully selected real-world industrial parts, supporting tools for scalable dataset expansion, diverse modalities, and ensured diversity in model complexity and geometric features, making it a comprehensive multimodal dataset built for semantic-driven CAD modeling and cross-modal learning.

Representing 3D surfaces as level sets of continuous functions over R^3 is the common denominator of neural implicit representations, which recently enabled remarkable progress in geometric deep learning and computer vision tasks. In order to represent 3D motion within this framework, it is often assumed (either explicitly or implicitly) that the transformations which a surface may undergo are homeomorphic: this is not necessarily true, for instance, in the case of fluid dynamics. In order to represent more general classes of deformations, we propose to apply this theoretical framework as regularizers for the optimization of simple 4D implicit functions (such as signed distance fields). We show that our representation is capable of capturing both homeomorphic and topology-changing deformations, while also defining correspondences over the continuously-reconstructed surfaces.

Vision-only occupancy prediction requires recovering a semantic 3D occupancy field from calibrated surround-view images, where each view provides observations with ambiguous depth along camera rays. Existing methods have progressed from dense structured representations to sparse Gaussian primitives, improving the efficiency of 3D scene representation. However, Gaussian learning still relies primarily on image domain features, which provide limited explicit geometric information for volumetric reasoning. Our key observation is that effective Gaussian occupancy modeling requires not only sparse primitives, but also richer geometric and semantic learning cues. In this paper, we propose VGOcc, which learns visual and geometric cues from foundation models for Gaussian modeling. VGOcc incorporates these cues into primitive initialization and refinement, yielding a representation termed Visual-Geometric Gaussians tailored to semantic occupancy prediction. Specifically, we propose Visual-Geometric Gaussian Birth to form spatially balanced Gaussian centers from ray depth hypotheses, while visual semantic features initialize primitive attributes. Next, we design Pose-Aware Feature Learning to combine foundation tokens with camera embeddings and calibrated ray information. Features from neighboring views are then aggregated at projected 3D locations for each Gaussian refinement stage. Finally, Gaussian decoder refines birth Gaussians with pose-aware features and renders them into semantic occupancy. Experiments on nuScenes demonstrate that VGOcc achieves state-of-the-art performance in vision-only 3D occupancy prediction. Codes will be available at https://github.com/JHLin42in/VGOcc.

Classifying the topology of closed curves is a central problem in low dimensional topology with applications beyond mathematics spanning protein folding, polymer physics and even magnetohydrodynamics. The central problem is how to determine whether two embeddings of a closed arc are equivalent under ambient isotopy. Given the striking ability of neural networks to solve complex classification tasks, it is therefore natural to ask if the knot classification problem can be tackled using Machine Learning (ML). In this paper, we investigate generic shortcut methods employed by ML to solve the knot classification challenge and specifically discover hidden non-topological features in training data generated through Molecular Dynamics simulations of polygonal knots that are used by ML to arrive to positive classifications results. We then provide a rigorous foundation for future attempts to tackle the knot classification challenge using ML by developing a publicly-available (i) dataset, that aims to remove the potential of non-topological feature classification and (ii) code, that can generate knot embeddings that faithfully explore chosen geometric state space with fixed knot topology. We expect that our work will accelerate the development of ML models that can solve complex geometric knot classification challenges.

Learning effective protein representations is critical in a variety of tasks in biology such as predicting protein function or structure. Existing approaches usually pretrain protein language models on a large number of unlabeled amino acid sequences and then finetune the models with some labeled data in downstream tasks. Despite the effectiveness of sequence-based approaches, the power of pretraining on known protein structures, which are available in smaller numbers only, has not been explored for protein property prediction, though protein structures are known to be determinants of protein function. In this paper, we propose to pretrain protein representations according to their 3D structures. We first present a simple yet effective encoder to learn the geometric features of a protein. We pretrain the protein graph encoder by leveraging multiview contrastive learning and different self-prediction tasks. Experimental results on both function prediction and fold classification tasks show that our proposed pretraining methods outperform or are on par with the state-of-the-art sequence-based methods, while using much less pretraining data. Our implementation is available at https://github.com/DeepGraphLearning/GearNet.

The aerodynamic coefficients of aircrafts are significantly impacted by its geometry, especially when the angle of attack (AoA) is large. In the field of aerodynamics, traditional polynomial-based parameterization uses as few parameters as possible to describe the geometry of an airfoil. However, because the 3D geometry of a wing is more complicated than the 2D airfoil, polynomial-based parameterizations have difficulty in accurately representing the entire shape of a wing in 3D space. Existing deep learning-based methods can extract massive latent neural representations for the shape of 2D airfoils or 2D slices of wings. Recent studies highlight that directly taking geometric features as inputs to the neural networks can improve the accuracy of predicted aerodynamic coefficients. Motivated by geometry theory, we propose to incorporate Riemannian geometric features for learning Coefficient of Pressure (CP) distributions on wing surfaces. Our method calculates geometric features (Riemannian metric, connection, and curvature) and further inputs the geometric features, coordinates and flight conditions into a deep learning model to predict the CP distribution. Experimental results show that our method, compared to state-of-the-art Deep Attention Network (DAN), reduces the predicted mean square error (MSE) of CP by an average of 8.41% for the DLR-F11 aircraft test set.

Synthetic data is crucial for advancing autonomous driving (AD) systems, yet current state-of-the-art video generation models, despite their visual realism, suffer from subtle geometric distortions that limit their utility for downstream perception tasks. We identify and quantify this critical issue, demonstrating a significant performance gap in 3D object detection when using synthetic versus real data. To address this, we introduce Reinforcement Learning with Geometric Feedback (RLGF), RLGF uniquely refines video diffusion models by incorporating rewards from specialized latent-space AD perception models. Its core components include an efficient Latent-Space Windowing Optimization technique for targeted feedback during diffusion, and a Hierarchical Geometric Reward (HGR) system providing multi-level rewards for point-line-plane alignment, and scene occupancy coherence. To quantify these distortions, we propose GeoScores. Applied to models like DiVE on nuScenes, RLGF substantially reduces geometric errors (e.g., VP error by 21\%, Depth error by 57\%) and dramatically improves 3D object detection mAP by 12.7\%, narrowing the gap to real-data performance. RLGF offers a plug-and-play solution for generating geometrically sound and reliable synthetic videos for AD development.

Representing shapes as level sets of neural networks has been recently proved to be useful for different shape analysis and reconstruction tasks. So far, such representations were computed using either: (i) pre-computed implicit shape representations; or (ii) loss functions explicitly defined over the neural level sets. In this paper we offer a new paradigm for computing high fidelity implicit neural representations directly from raw data (i.e., point clouds, with or without normal information). We observe that a rather simple loss function, encouraging the neural network to vanish on the input point cloud and to have a unit norm gradient, possesses an implicit geometric regularization property that favors smooth and natural zero level set surfaces, avoiding bad zero-loss solutions. We provide a theoretical analysis of this property for the linear case, and show that, in practice, our method leads to state of the art implicit neural representations with higher level-of-details and fidelity compared to previous methods.

WiFi sensing technology has shown superiority in smart homes among various sensors for its cost-effective and privacy-preserving merits. It is empowered by Channel State Information (CSI) extracted from WiFi signals and advanced machine learning models to analyze motion patterns in CSI. Many learning-based models have been proposed for kinds of applications, but they severely suffer from environmental dependency. Though domain adaptation methods have been proposed to tackle this issue, it is not practical to collect high-quality, well-segmented and balanced CSI samples in a new environment for adaptation algorithms, but randomly-captured CSI samples can be easily collected. {blackIn this paper, we firstly explore how to learn a robust model from these low-quality CSI samples, and propose AutoFi, an annotation-efficient WiFi sensing model based on a novel geometric self-supervised learning algorithm.} The AutoFi fully utilizes unlabeled low-quality CSI samples that are captured randomly, and then transfers the knowledge to specific tasks defined by users, which is the first work to achieve cross-task transfer in WiFi sensing. The AutoFi is implemented on a pair of Atheros WiFi APs for evaluation. The AutoFi transfers knowledge from randomly collected CSI samples into human gait recognition and achieves state-of-the-art performance. Furthermore, we simulate cross-task transfer using public datasets to further demonstrate its capacity for cross-task learning. For the UT-HAR and Widar datasets, the AutoFi achieves satisfactory results on activity recognition and gesture recognition without any prior training. We believe that the AutoFi takes a huge step toward automatic WiFi sensing without any developer engagement.

Shape assembly aims to reassemble parts (or fragments) into a complete object, which is a common task in our daily life. Different from the semantic part assembly (e.g., assembling a chair's semantic parts like legs into a whole chair), geometric part assembly (e.g., assembling bowl fragments into a complete bowl) is an emerging task in computer vision and robotics. Instead of semantic information, this task focuses on geometric information of parts. As the both geometric and pose space of fractured parts are exceptionally large, shape pose disentanglement of part representations is beneficial to geometric shape assembly. In our paper, we propose to leverage SE(3) equivariance for such shape pose disentanglement. Moreover, while previous works in vision and robotics only consider SE(3) equivariance for the representations of single objects, we move a step forward and propose leveraging SE(3) equivariance for representations considering multi-part correlations, which further boosts the performance of the multi-part assembly. Experiments demonstrate the significance of SE(3) equivariance and our proposed method for geometric shape assembly. Project page: https://crtie.github.io/SE-3-part-assembly/

The integration of large language models (LLMs) into medical practice holds transformative potential, yet their real-world clinical utility remains limited by critical alignment challenges: (1) a disconnect between static evaluation benchmarks and dynamic clinical cognitive needs, (2) difficulties in adapting to evolving, multi-source medical standards, and (3) the inability of conventional reward models to capture nuanced, multi-dimensional medical quality criteria. To address these gaps, we propose MR-RML (Multidimensional Rubric-oriented Reward Model Learning) via GPRC (Geometric Projection Reference Constraints), a novel alignment framework that integrates medical standards into a structured "Dimensions-Scenarios-Disciplines" matrix to guide data generation and model optimization. MR-RML introduces three core innovations: (1) a "Dimensions-Scenarios-Disciplines" medical standard system that embeds domain standards into the full training pipeline; (2) an independent multi-dimensional reward model that decomposes evaluation criteria, shifting from real-time rubric-based scoring to internalized reward modeling for improved consistency and cost-efficiency; (3) geometric projection reference constraints that transform medical cognitive logic into mathematical regularization, aligning scoring gradients with clinical reasoning and enabling synthetic data-driven training. Through extensive evaluations on the authoritative medical benchmark Healthbench, our method yields substantial performance gains over the base LLM Qwen-32B (45% on the full subset and 85% on Hard subset, respectively). It achieves a SOTA among open-source LLMs with scores of 62.7 (full subset) and 44.7 (hard subset), while also outperforming the majority of closed-source models.

The detection of small road hazards, such as lost cargo, is a vital capability for self-driving cars. We tackle this challenging and rarely addressed problem with a vision system that leverages appearance, contextual as well as geometric cues. To utilize the appearance and contextual cues, we propose a new deep learning-based obstacle detection framework. Here a variant of a fully convolutional network is used to predict a pixel-wise semantic labeling of (i) free-space, (ii) on-road unexpected obstacles, and (iii) background. The geometric cues are exploited using a state-of-the-art detection approach that predicts obstacles from stereo input images via model-based statistical hypothesis tests. We present a principled Bayesian framework to fuse the semantic and stereo-based detection results. The mid-level Stixel representation is used to describe obstacles in a flexible, compact and robust manner. We evaluate our new obstacle detection system on the Lost and Found dataset, which includes very challenging scenes with obstacles of only 5 cm height. Overall, we report a major improvement over the state-of-the-art, with relative performance gains of up to 50%. In particular, we achieve a detection rate of over 90% for distances of up to 50 m. Our system operates at 22 Hz on our self-driving platform.

Category-level 6D object pose estimation aims to estimate the rotation, translation and size of unseen instances within specific categories. In this area, dense correspondence-based methods have achieved leading performance. However, they do not explicitly consider the local and global geometric information of different instances, resulting in poor generalization ability to unseen instances with significant shape variations. To deal with this problem, we propose a novel Instance-Adaptive and Geometric-Aware Keypoint Learning method for category-level 6D object pose estimation (AG-Pose), which includes two key designs: (1) The first design is an Instance-Adaptive Keypoint Detection module, which can adaptively detect a set of sparse keypoints for various instances to represent their geometric structures. (2) The second design is a Geometric-Aware Feature Aggregation module, which can efficiently integrate the local and global geometric information into keypoint features. These two modules can work together to establish robust keypoint-level correspondences for unseen instances, thus enhancing the generalization ability of the model.Experimental results on CAMERA25 and REAL275 datasets show that the proposed AG-Pose outperforms state-of-the-art methods by a large margin without category-specific shape priors.

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently nonEuclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.

A key challenge in machine learning is to explain how learning dynamics select among the many solutions that achieve identical loss values in overparameterized models - a phenomenon known as implicit bias. Controlling this bias provides a direct mechanism on learned representations, which are central to interpretability, robustness, and reasoning in modern AI systems. Yet, despite its importance, existing explanations remain largely ad hoc and lack a unifying mechanism. We develop a theoretical and constructive framework in which implicit bias emerges as a geometric correction induced by the interplay between gradient noise and continuous symmetries of the loss. We compute the induced bias across a range of architectures, predicting new behaviors and explaining known ones. The approach also enables inverse design: by engineering predictor - preserving parameterizations, it is possible to shape the bias, with sparsity and spectral sparsity emerging as canonical instances. Numerical experiments support the theory and validate the inverse - design framework in controlled settings.

In this work, we present a novel learning-based framework that combines the local accuracy of contrastive learning with the global consistency of geometric approaches, for robust non-rigid matching. We first observe that while contrastive learning can lead to powerful point-wise features, the learned correspondences commonly lack smoothness and consistency, owing to the purely combinatorial nature of the standard contrastive losses. To overcome this limitation we propose to boost contrastive feature learning with two types of smoothness regularization that inject geometric information into correspondence learning. With this novel combination in hand, the resulting features are both highly discriminative across individual points, and, at the same time, lead to robust and consistent correspondences, through simple proximity queries. Our framework is general and is applicable to local feature learning in both the 3D and 2D domains. We demonstrate the superiority of our approach through extensive experiments on a wide range of challenging matching benchmarks, including 3D non-rigid shape correspondence and 2D image keypoint matching.

Geometric methods for solving open-world off-road navigation tasks, by learning occupancy and metric maps, provide good generalization but can be brittle in outdoor environments that violate their assumptions (e.g., tall grass). Learning-based methods can directly learn collision-free behavior from raw observations, but are difficult to integrate with standard geometry-based pipelines. This creates an unfortunate conflict -- either use learning and lose out on well-understood geometric navigational components, or do not use it, in favor of extensively hand-tuned geometry-based cost maps. In this work, we reject this dichotomy by designing the learning and non-learning-based components in a way such that they can be effectively combined in a self-supervised manner. Both components contribute to a planning criterion: the learned component contributes predicted traversability as rewards, while the geometric component contributes obstacle cost information. We instantiate and comparatively evaluate our system in both in-distribution and out-of-distribution environments, showing that this approach inherits complementary gains from the learned and geometric components and significantly outperforms either of them. Videos of our results are hosted at https://sites.google.com/view/hybrid-imitative-planning

We introduce Point2Skeleton, an unsupervised method to learn skeletal representations from point clouds. Existing skeletonization methods are limited to tubular shapes and the stringent requirement of watertight input, while our method aims to produce more generalized skeletal representations for complex structures and handle point clouds. Our key idea is to use the insights of the medial axis transform (MAT) to capture the intrinsic geometric and topological natures of the original input points. We first predict a set of skeletal points by learning a geometric transformation, and then analyze the connectivity of the skeletal points to form skeletal mesh structures. Extensive evaluations and comparisons show our method has superior performance and robustness. The learned skeletal representation will benefit several unsupervised tasks for point clouds, such as surface reconstruction and segmentation.

Auto-Regressive (AR) models have achieved impressive results in 2D image generation by modeling joint distributions in the grid space. While this approach has been extended to the 3D domain for powerful shape generation, it still has two limitations: expensive computations on volumetric grids and ambiguous auto-regressive order along grid dimensions. To overcome these limitations, we propose the Improved Auto-regressive Model (ImAM) for 3D shape generation, which applies discrete representation learning based on a latent vector instead of volumetric grids. Our approach not only reduces computational costs but also preserves essential geometric details by learning the joint distribution in a more tractable order. Moreover, thanks to the simplicity of our model architecture, we can naturally extend it from unconditional to conditional generation by concatenating various conditioning inputs, such as point clouds, categories, images, and texts. Extensive experiments demonstrate that ImAM can synthesize diverse and faithful shapes of multiple categories, achieving state-of-the-art performance.

We investigate the geometric structure of learning dynamics in overparameterized transformer models through carefully controlled modular arithmetic tasks. Our primary finding is that despite operating in high-dimensional parameter spaces (d=128), transformer training trajectories rapidly collapse onto low-dimensional execution manifolds of dimension 3--4. This dimensional collapse is robust across random seeds and moderate task difficulties, though the orientation of the manifold in parameter space varies between runs. We demonstrate that this geometric structure underlies several empirically observed phenomena: (1) sharp attention concentration emerges as saturation along routing coordinates within the execution manifold, (2) SGD commutators are preferentially aligned with the execution subspace (up to 10times random baseline) early in training, with >92% of non-commutativity confined to orthogonal staging directions and this alignment decreasing as training converges, and (3) sparse autoencoders capture auxiliary routing structure but fail to isolate execution itself, which remains distributed across the low-dimensional manifold. Our results suggest a unifying geometric framework for understanding transformer learning, where the vast majority of parameters serve to absorb optimization interference while core computation occurs in a dramatically reduced subspace. These findings have implications for interpretability, training curriculum design, and understanding the role of overparameterization in neural network learning.

Generating structured, editable diagrams remains a significant challenge for contemporary large language models, despite their proficiency in general-purpose vector code generation. The primary difficulty lies in the structural fragility of the output; minor errors such as misaligned connector endpoints, text labels overlapping borders, or complex layouts drifting beyond the canvas boundaries render the resulting SVG files functionally unusable for professional applications. To address these issues, we introduce GeoSVG-RL, a specialized reinforcement learning framework designed for layout-constrained text-to-SVG generation. Unlike standard training objectives that rely solely on maximizing token-level likelihood, our approach optimizes the policy against explicit, executable geometric feedback. The model first produces a structured layout plan that serves as a geometric contract for the subsequent generation of the SVG code. This code is then rendered through a browser-backed verifier, enabling the calculation of fine-grained rewards across six critical dimensions: rendering validity, canvas fitting, precise anchor placement, text containment, graph consistency, and code cleanliness. We utilize Group Relative Policy Optimization (GRPO) to refine the model, sampling multiple candidates per prompt to facilitate updates based on relative quality. Starting from a supervised warm-start phase on synthetic data, GeoSVG-RL achieves substantial gains in structural reliability, particularly in arrow-anchor accuracy and text-in-box rates. Quantitative evaluations demonstrate that our method consistently outperforms current state-of-the-art systems in local geometric precision and the preservation of graph connectivity, providing a robust pathway toward automated yet reliable technical illustration.

Dense contrastive representation learning (DCRL) has greatly improved the learning efficiency for image-dense prediction tasks, showing its great potential to reduce the large costs of medical image collection and dense annotation. However, the properties of medical images make unreliable correspondence discovery, bringing an open problem of large-scale false positive and negative (FP&N) pairs in DCRL. In this paper, we propose GEoMetric vIsual deNse sImilarity (GEMINI) learning which embeds the homeomorphism prior to DCRL and enables a reliable correspondence discovery for effective dense contrast. We propose a deformable homeomorphism learning (DHL) which models the homeomorphism of medical images and learns to estimate a deformable mapping to predict the pixels' correspondence under topological preservation. It effectively reduces the searching space of pairing and drives an implicit and soft learning of negative pairs via a gradient. We also propose a geometric semantic similarity (GSS) which extracts semantic information in features to measure the alignment degree for the correspondence learning. It will promote the learning efficiency and performance of deformation, constructing positive pairs reliably. We implement two practical variants on two typical representation learning tasks in our experiments. Our promising results on seven datasets which outperform the existing methods show our great superiority. We will release our code on a companion link: https://github.com/YutingHe-list/GEMINI.

We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised learning. Additionally, unsupervised methods can provide insight into the structure of such geometrical data. At the heart of this programme is the question of how geometry can be machine learned, and indeed how AI helps one to do mathematics. This is a chapter contribution to the book Machine learning and Algebraic Geometry, edited by A. Kasprzyk et al.

Machine learning techniques are essential tools to compute efficient, yet accurate, force fields for atomistic simulations. This approach has recently been extended to incorporate quantum computational methods, making use of variational quantum learning models to predict potential energy surfaces and atomic forces from ab initio training data. However, the trainability and scalability of such models are still limited, due to both theoretical and practical barriers. Inspired by recent developments in geometric classical and quantum machine learning, here we design quantum neural networks that explicitly incorporate, as a data-inspired prior, an extensive set of physically relevant symmetries. We find that our invariant quantum learning models outperform their more generic counterparts on individual molecules of growing complexity. Furthermore, we study a water dimer as a minimal example of a system with multiple components, showcasing the versatility of our proposed approach and opening the way towards larger simulations. Our results suggest that molecular force fields generation can significantly profit from leveraging the framework of geometric quantum machine learning, and that chemical systems represent, in fact, an interesting and rich playground for the development and application of advanced quantum machine learning tools.

Vision-Language Models (VLMs) often struggle with robust 3D spatial reasoning. Prevailing methods that rely on fine-tuning with 3D visual question-answering (VQA) datasets may overfit dataset-specific biases, while integrating specialized 3D visual encoders is often inflexible and cumbersome. In this paper, we argue that genuine spatial understanding should emerge from learning fundamental geometric priors, not only from high-level VQA supervision. We propose GASP (Geometric-Aware Spatial Priors), a framework that injects these priors directly into the LLM's transformer layers. GASP employs a small correspondence head, applied as a deep supervision signal across all layers, and is trained with a dual objective leveraging ground-truth geometry from large-scale video scenes: a contrastive loss on ground-truth point correspondences enforces 2D view-invariance, while a depth consistency supervision resolves 3D geometric ambiguities. Our analysis first provides a diagnostic showing that standard VLMs' internal correspondence matching accuracy is very low (often below 5%). We then demonstrate that our training substantially improves this behavior, boosting peak layer-wise correspondence to over 70% and maintaining over 85% temporal robustness while baselines remain below 5%. These internal improvements translate to significant gains on downstream spatial benchmarks including +18.2% on All-Angles Bench and +29.0% on VSI-Bench, all without training on any 3D VQA data. Our findings indicate that learning from fundamental geometric priors is a promising and generalizable pathway towards VLMs with more reliable 3D spatial reasoning.

Unsupervised learning for geometric perception (depth, optical flow, etc.) is of great interest to autonomous systems. Recent works on unsupervised learning have made considerable progress on perceiving geometry; however, they usually ignore the coherence of objects and perform poorly under scenarios with dark and noisy environments. In contrast, supervised learning algorithms, which are robust, require large labeled geometric dataset. This paper introduces SIGNet, a novel framework that provides robust geometry perception without requiring geometrically informative labels. Specifically, SIGNet integrates semantic information to make depth and flow predictions consistent with objects and robust to low lighting conditions. SIGNet is shown to improve upon the state-of-the-art unsupervised learning for depth prediction by 30% (in squared relative error). In particular, SIGNet improves the dynamic object class performance by 39% in depth prediction and 29% in flow prediction. Our code will be made available at https://github.com/mengyuest/SIGNet

Variational algorithms require architectures that naturally constrain the optimisation space to run efficiently. In geometric quantum machine learning, one achieves this by encoding group structure into parameterised quantum circuits to include the symmetries of a problem as an inductive bias. However, constructing such circuits is challenging as a concrete guiding principle has yet to emerge. In this paper, we propose the use of spin networks, a form of directed tensor network invariant under a group transformation, to devise SU(2) equivariant quantum circuit ans\"atze -- circuits possessing spin rotation symmetry. By changing to the basis that block diagonalises SU(2) group action, these networks provide a natural building block for constructing parameterised equivariant quantum circuits. We prove that our construction is mathematically equivalent to other known constructions, such as those based on twirling and generalised permutations, but more direct to implement on quantum hardware. The efficacy of our constructed circuits is tested by solving the ground state problem of SU(2) symmetric Heisenberg models on the one-dimensional triangular lattice and on the Kagome lattice. Our results highlight that our equivariant circuits boost the performance of quantum variational algorithms, indicating broader applicability to other real-world problems.

3D Gaussians have recently emerged as an effective scene representation for real-time splatting and accurate novel-view synthesis, motivating several works to adapt multi-view structure prediction networks to regress per-pixel 3D Gaussians from images. However, most prior work extends these networks to predict additional Gaussian parameters -- orientation, scale, opacity, and appearance -- while relying almost exclusively on view-synthesis supervision. We show that a view-synthesis loss alone is insufficient to recover geometrically meaningful splats in this setting. We analyze and address the ambiguities of learning 3D Gaussian splats under self-supervision for pose-free generalizable splatting, and introduce G3Splat, which enforces geometric priors to obtain geometrically consistent 3D scene representations. Trained on RE10K, our approach achieves state-of-the-art performance in (i) geometrically consistent reconstruction, (ii) relative pose estimation, and (iii) novel-view synthesis. We further demonstrate strong zero-shot generalization on ScanNet, substantially outperforming prior work in both geometry recovery and relative pose estimation. Code and pretrained models are released on our project page (https://m80hz.github.io/g3splat/).

Training recurrent neural networks (RNNs) is a high-dimensional process that requires updating numerous parameters. Therefore, it is often difficult to pinpoint the underlying learning mechanisms. To address this challenge, we propose to gain mechanistic insights into the phenomenon of abrupt learning by studying RNNs trained to perform diverse short-term memory tasks. In these tasks, RNN training begins with an initial search phase. Following a long period of plateau in accuracy, the values of the loss function suddenly drop, indicating abrupt learning. Analyzing the neural computation performed by these RNNs reveals geometric restructuring (GR) in their phase spaces prior to the drop. To promote these GR events, we introduce a temporal consistency regularization that accelerates (bioplausible) training, facilitates attractor formation, and enables efficient learning in strongly connected networks. Our findings offer testable predictions for neuroscientists and emphasize the need for goal-agnostic secondary mechanisms to facilitate learning in biological and artificial networks.

Solving the camera-to-robot pose is a fundamental requirement for vision-based robot control, and is a process that takes considerable effort and cares to make accurate. Traditional approaches require modification of the robot via markers, and subsequent deep learning approaches enabled markerless feature extraction. Mainstream deep learning methods only use synthetic data and rely on Domain Randomization to fill the sim-to-real gap, because acquiring the 3D annotation is labor-intensive. In this work, we go beyond the limitation of 3D annotations for real-world data. We propose an end-to-end pose estimation framework that is capable of online camera-to-robot calibration and a self-supervised training method to scale the training to unlabeled real-world data. Our framework combines deep learning and geometric vision for solving the robot pose, and the pipeline is fully differentiable. To train the Camera-to-Robot Pose Estimation Network (CtRNet), we leverage foreground segmentation and differentiable rendering for image-level self-supervision. The pose prediction is visualized through a renderer and the image loss with the input image is back-propagated to train the neural network. Our experimental results on two public real datasets confirm the effectiveness of our approach over existing works. We also integrate our framework into a visual servoing system to demonstrate the promise of real-time precise robot pose estimation for automation tasks.

Many existing learning-based grasping approaches concentrate on a single embodiment, provide limited generalization to higher DoF end-effectors and cannot capture a diverse set of grasp modes. We tackle the problem of grasping using multiple embodiments by learning rich geometric representations for both objects and end-effectors using Graph Neural Networks. Our novel method - GeoMatch - applies supervised learning on grasping data from multiple embodiments, learning end-to-end contact point likelihood maps as well as conditional autoregressive predictions of grasps keypoint-by-keypoint. We compare our method against baselines that support multiple embodiments. Our approach performs better across three end-effectors, while also producing diverse grasps. Examples, including real robot demos, can be found at geo-match.github.io.

Crystal structures are characterized by atomic bases within a primitive unit cell that repeats along a regular lattice throughout 3D space. The periodic and infinite nature of crystals poses unique challenges for geometric graph representation learning. Specifically, constructing graphs that effectively capture the complete geometric information of crystals and handle chiral crystals remains an unsolved and challenging problem. In this paper, we introduce a novel approach that utilizes the periodic patterns of unit cells to establish the lattice-based representation for each atom, enabling efficient and expressive graph representations of crystals. Furthermore, we propose ComFormer, a SE(3) transformer designed specifically for crystalline materials. ComFormer includes two variants; namely, iComFormer that employs invariant geometric descriptors of Euclidean distances and angles, and eComFormer that utilizes equivariant vector representations. Experimental results demonstrate the state-of-the-art predictive accuracy of ComFormer variants on various tasks across three widely-used crystal benchmarks. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

The choice of data representation is a key factor in the success of deep learning in geometric tasks. For instance, DUSt3R recently introduced the concept of viewpoint-invariant point maps, generalizing depth prediction and showing that all key problems in the 3D reconstruction of static scenes can be reduced to predicting such point maps. In this paper, we develop an analogous concept for a very different problem: the reconstruction of the 3D shape and pose of deformable objects. To this end, we introduce Dual Point Maps (DualPM), where a pair of point maps is extracted from the same image-one associating pixels to their 3D locations on the object and the other to a canonical version of the object in its rest pose. We also extend point maps to amodal reconstruction to recover the complete shape of the object, even through self-occlusions. We show that 3D reconstruction and 3D pose estimation can be reduced to the prediction of DualPMs. Empirically, we demonstrate that this representation is a suitable target for deep networks to predict. Specifically, we focus on modeling quadrupeds, showing that DualPMs can be trained purely on synthetic 3D data, consisting of one or two models per category, while generalizing effectively to real images. With this approach, we achieve significant improvements over previous methods for the 3D analysis and reconstruction of such objects.

Auto-regressive models have achieved impressive results in 2D image generation by modeling joint distributions in grid space. In this paper, we extend auto-regressive models to 3D domains, and seek a stronger ability of 3D shape generation by improving auto-regressive models at capacity and scalability simultaneously. Firstly, we leverage an ensemble of publicly available 3D datasets to facilitate the training of large-scale models. It consists of a comprehensive collection of approximately 900,000 objects, with multiple properties of meshes, points, voxels, rendered images, and text captions. This diverse labeled dataset, termed Objaverse-Mix, empowers our model to learn from a wide range of object variations. However, directly applying 3D auto-regression encounters critical challenges of high computational demands on volumetric grids and ambiguous auto-regressive order along grid dimensions, resulting in inferior quality of 3D shapes. To this end, we then present a novel framework Argus3D in terms of capacity. Concretely, our approach introduces discrete representation learning based on a latent vector instead of volumetric grids, which not only reduces computational costs but also preserves essential geometric details by learning the joint distributions in a more tractable order. The capacity of conditional generation can thus be realized by simply concatenating various conditioning inputs to the latent vector, such as point clouds, categories, images, and texts. In addition, thanks to the simplicity of our model architecture, we naturally scale up our approach to a larger model with an impressive 3.6 billion parameters, further enhancing the quality of versatile 3D generation. Extensive experiments on four generation tasks demonstrate that Argus3D can synthesize diverse and faithful shapes across multiple categories, achieving remarkable performance.

We present FLARE, a feed-forward model designed to infer high-quality camera poses and 3D geometry from uncalibrated sparse-view images (i.e., as few as 2-8 inputs), which is a challenging yet practical setting in real-world applications. Our solution features a cascaded learning paradigm with camera pose serving as the critical bridge, recognizing its essential role in mapping 3D structures onto 2D image planes. Concretely, FLARE starts with camera pose estimation, whose results condition the subsequent learning of geometric structure and appearance, optimized through the objectives of geometry reconstruction and novel-view synthesis. Utilizing large-scale public datasets for training, our method delivers state-of-the-art performance in the tasks of pose estimation, geometry reconstruction, and novel view synthesis, while maintaining the inference efficiency (i.e., less than 0.5 seconds). The project page and code can be found at: https://zhanghe3z.github.io/FLARE/

We introduce a generalized attention mechanism for spherical domains, enabling Transformer architectures to natively process data defined on the two-dimensional sphere - a critical need in fields such as atmospheric physics, cosmology, and robotics, where preserving spherical symmetries and topology is essential for physical accuracy. By integrating numerical quadrature weights into the attention mechanism, we obtain a geometrically faithful spherical attention that is approximately rotationally equivariant, providing strong inductive biases and leading to better performance than Cartesian approaches. To further enhance both scalability and model performance, we propose neighborhood attention on the sphere, which confines interactions to geodesic neighborhoods. This approach reduces computational complexity and introduces the additional inductive bias for locality, while retaining the symmetry properties of our method. We provide optimized CUDA kernels and memory-efficient implementations to ensure practical applicability. The method is validated on three diverse tasks: simulating shallow water equations on the rotating sphere, spherical image segmentation, and spherical depth estimation. Across all tasks, our spherical Transformers consistently outperform their planar counterparts, highlighting the advantage of geometric priors for learning on spherical domains.

Recent advances in deep learning for remote sensing rely heavily on large annotated datasets, yet acquiring high-quality ground truth for geometric, radiometric, and multi-domain tasks remains costly and often infeasible. In particular, the lack of accurate depth annotations, controlled illumination variations, and multi-scale paired imagery limits progress in monocular depth estimation, domain adaptation, and super-resolution for aerial scenes. We present SyMTRS, a large-scale synthetic dataset generated using a high-fidelity urban simulation pipeline. The dataset provides high-resolution RGB aerial imagery (2048 x 2048), pixel-perfect depth maps, night-time counterparts for domain adaptation, and aligned low-resolution variants for super-resolution at x2, x4, and x8 scales. Unlike existing remote sensing datasets that focus on a single task or modality, SyMTRS is designed as a unified multi-task benchmark enabling joint research in geometric understanding, cross-domain robustness, and resolution enhancement. We describe the dataset generation process, its statistical properties, and its positioning relative to existing benchmarks. SyMTRS aims to bridge critical gaps in remote sensing research by enabling controlled experiments with perfect geometric ground truth and consistent multi-domain supervision. The results obtained in this work can be reproduced from this Github repository: https://github.com/safouaneelg/SyMTRS.

Depth completion involves predicting dense depth maps from sparse LiDAR inputs. However, sparse depth annotations from sensors limit the availability of dense supervision, which is necessary for learning detailed geometric features. In this paper, we propose a two-stage knowledge distillation framework that leverages powerful monocular foundation models to provide dense supervision for depth completion. In the first stage, we introduce a pre-training strategy that generates diverse training data from natural images, which distills geometric knowledge to depth completion. Specifically, we simulate LiDAR scans by utilizing monocular depth and mesh reconstruction, thereby creating training data without requiring ground-truth depth. Besides, monocular depth estimation suffers from inherent scale ambiguity in real-world settings. To address this, in the second stage, we employ a scale- and shift-invariant loss (SSI Loss) to learn real-world scales when fine-tuning on real-world datasets. Our two-stage distillation framework enables depth completion models to harness the strengths of monocular foundation models. Experimental results demonstrate that models trained with our two-stage distillation framework achieve state-of-the-art performance, ranking first place on the KITTI benchmark. Code is available at https://github.com/Sharpiless/DMD3C

In machine learning, there is a long history of trying to build neural networks that can learn from fewer example data by baking in strong geometric priors. However, it is not always clear a priori what geometric constraints are appropriate for a given task. Here, we explore the possibility that one can uncover useful geometric inductive biases by studying how training molds the Riemannian geometry induced by unconstrained neural network feature maps. We first show that at infinite width, neural networks with random parameters induce highly symmetric metrics on input space. This symmetry is broken by feature learning: networks trained to perform classification tasks learn to magnify local areas along decision boundaries. This holds in deep networks trained on high-dimensional image classification tasks, and even in self-supervised representation learning. These results begin to elucidate how training shapes the geometry induced by unconstrained neural network feature maps, laying the groundwork for an understanding of this richly nonlinear form of feature learning.

Deep learning-based object detection and instance segmentation have achieved unprecedented progress. In this paper, we propose Complete-IoU (CIoU) loss and Cluster-NMS for enhancing geometric factors in both bounding box regression and Non-Maximum Suppression (NMS), leading to notable gains of average precision (AP) and average recall (AR), without the sacrifice of inference efficiency. In particular, we consider three geometric factors, i.e., overlap area, normalized central point distance and aspect ratio, which are crucial for measuring bounding box regression in object detection and instance segmentation. The three geometric factors are then incorporated into CIoU loss for better distinguishing difficult regression cases. The training of deep models using CIoU loss results in consistent AP and AR improvements in comparison to widely adopted ell_n-norm loss and IoU-based loss. Furthermore, we propose Cluster-NMS, where NMS during inference is done by implicitly clustering detected boxes and usually requires less iterations. Cluster-NMS is very efficient due to its pure GPU implementation, and geometric factors can be incorporated to improve both AP and AR. In the experiments, CIoU loss and Cluster-NMS have been applied to state-of-the-art instance segmentation (e.g., YOLACT and BlendMask-RT), and object detection (e.g., YOLO v3, SSD and Faster R-CNN) models. Taking YOLACT on MS COCO as an example, our method achieves performance gains as +1.7 AP and +6.2 AR_{100} for object detection, and +0.9 AP and +3.5 AR_{100} for instance segmentation, with 27.1 FPS on one NVIDIA GTX 1080Ti GPU. All the source code and trained models are available at https://github.com/Zzh-tju/CIoU

Neural surface representation has demonstrated remarkable success in the areas of novel view synthesis and 3D reconstruction. However, assessing the geometric quality of 3D reconstructions in the absence of ground truth mesh remains a significant challenge, due to its rendering-based optimization process and entangled learning of appearance and geometry with photometric losses. In this paper, we present a novel framework, i.e, GURecon, which establishes a geometric uncertainty field for the neural surface based on geometric consistency. Different from existing methods that rely on rendering-based measurement, GURecon models a continuous 3D uncertainty field for the reconstructed surface, and is learned by an online distillation approach without introducing real geometric information for supervision. Moreover, in order to mitigate the interference of illumination on geometric consistency, a decoupled field is learned and exploited to finetune the uncertainty field. Experiments on various datasets demonstrate the superiority of GURecon in modeling 3D geometric uncertainty, as well as its plug-and-play extension to various neural surface representations and improvement on downstream tasks such as incremental reconstruction. The code and supplementary material are available on the project website: https://zju3dv.github.io/GURecon/.

Data heterogeneity in federated learning, characterized by a significant misalignment between local and global distributions, leads to divergent local optimization directions and hinders global model training. Existing studies mainly focus on optimizing local updates or global aggregation, but these indirect approaches demonstrate instability when handling highly heterogeneous data distributions, especially in scenarios where label skew and domain skew coexist. To address this, we propose a geometry-guided data generation method that centers on simulating the global embedding distribution locally. We first introduce the concept of the geometric shape of an embedding distribution and then address the challenge of obtaining global geometric shapes under privacy constraints. Subsequently, we propose GGEUR, which leverages global geometric shapes to guide the generation of new samples, enabling a closer approximation to the ideal global distribution. In single-domain scenarios, we augment samples based on global geometric shapes to enhance model generalization; in multi-domain scenarios, we further employ class prototypes to simulate the global distribution across domains. Extensive experimental results demonstrate that our method significantly enhances the performance of existing approaches in handling highly heterogeneous data, including scenarios with label skew, domain skew, and their coexistence. Code published at: https://github.com/WeiDai-David/2025CVPR_GGEUR

PGD adversarial training, the standard robustness method, can reduce Jacobian Frobenius norm yet worsen clean-input geometry (e.g., TDI 1.336 vs. ERM 1.093). We show this is not an implementation artifact but a theorem-level consequence of supervised learning. We prove that any encoder minimizing supervised loss must retain non-zero sensitivity along directions correlated with training labels, including directions that are nuisance at test time. This holds across proper scoring rules, architectures, and dataset sizes. We call this the geometric blind spot of supervised learning. This theorem unifies four empirical phenomena often treated separately: non-robust features, texture bias, corruption fragility, and the robustness-accuracy tradeoff. It also explains why suppressing sensitivity in one adversarial direction can redistribute sensitivity elsewhere. We introduce Trajectory Deviation Index (TDI), a diagnostic of geometric isotropy. Unlike CKA, intrinsic dimension, or Jacobian Frobenius norm alone, TDI captures the failure mode above. In our experiments, PGD attains low Frobenius norm but high TDI, while PMH attains the lowest TDI with one additional training term and no architectural changes. Across seven tasks, BERT/SST-2, and ImageNet ViT-B/16 (backbone family underlying CLIP/DINO/SAM), the blind spot is measurable and repairable. It appears at foundation-model scale, worsens with model scale and task-specific fine-tuning, and is substantially reduced by PMH. PMH also leads on non-Gaussian corruption types (blur/brightness/contrast) without corruption-specific training.

Learning visuomotor policies from scarce expert demonstrations remains a core challenge in robotic manipulation. A primary hurdle lies in distilling high-dimensional RGB representations into control-relevant geometry without overfitting. While using frozen pre-trained Vision Foundation Models (VFMs) improves data efficiency, it also shifts most task adaptation onto a small spatial pooling module, which can latch onto task-irrelevant shortcuts and lose geometric grounding when finetuned with few data samples. More broadly, pre-trained visual representations used for policy learning have been observed to struggle under even minor scene perturbations, highlighting the need for robustness-oriented inductive biases. We propose Geometric Anchor Pre-training (GAP), a simple, action-free warm-up stage that regularizes the spatial adapter before downstream imitation learning. GAP pre-trains the pooling layer on a lightweight simulated proxy task where object masks are available at no cost, encouraging the adapter to produce keypoints that lie on the object, cover its spatial extent, and remain sharp and repeatable over time. This yields stable geometric anchors that provide a reliable coordinate interface for few-shot policy learning, while keeping the VFM frozen. We evaluate GAP on RoboMimic and ManiSkill under severe data scarcity (15-50 demonstrations) and domain shift. A simple adapter regularized with GAP consistently outperforms stronger attention-based poolers and end-to-end fine-tuning, achieving 62% success on RoboMimic Can with 15 demonstrations (+16% over AFA), 63% on the long-horizon high-precision Tool Hang task with 50 demonstrations, and 61% on ManiSkill StackCube with 30 demonstrations (+11% over full fine-tuning). The proxy stage is lightweight and fully decoupled from downstream tasks, making it practical to reuse across environments and manipulation skills.

Semi-supervised learning has emerged as a powerful paradigm for leveraging large amounts of unlabeled data to improve the performance of machine learning models when labeled data are scarce. Among existing approaches, methods derived from FixMatch have achieved state-of-the-art results in image classification by combining weak and strong data augmentations with confidence-based pseudo-labeling. Despite their strong empirical performance, these methods typically struggle with two critical bottlenecks: majority classes tend to dominate the learning process, which is amplified by incorrect pseudo-labels, leading to biased models. Furthermore, noisy early pseudo-labels prevent the model from forming clear decision boundaries, requiring prolonged training to learn informative representation. In this paper, we introduce a paradigm shift from conventional logical output threshold base, toward an explicit shaping of geometric representations. Our approach is inspired by the recently proposed Latent-Euclidean Joint-Embedding Predictive Architectures (LeJEPA), a theoretically grounded framework asserting that meaningful representations should exhibit an isotropic Gaussian structure in latent space. Building on this principle, we propose a new training objective that combines the classical semi-supervised loss used in FlexMatch, an adaptive extension of FixMatch, with a latent-space regularization term derived from LeJEPA. Our proposed approach, encourages well-structured representations while preserving the advantages of pseudo-labeling strategies. Through extensive experiments on CIFAR-100, STL-10 and Tiny-ImageNet, we demonstrate that the proposed method consistently outperforms existing baselines. In addition, our method significantly accelerates the convergence, drastically reducing the overall computational cost compared to standard FixMatch-based pipelines.

From a single image, visual cues can help deduce intrinsic and extrinsic camera parameters like the focal length and the gravity direction. This single-image calibration can benefit various downstream applications like image editing and 3D mapping. Current approaches to this problem are based on either classical geometry with lines and vanishing points or on deep neural networks trained end-to-end. The learned approaches are more robust but struggle to generalize to new environments and are less accurate than their classical counterparts. We hypothesize that they lack the constraints that 3D geometry provides. In this work, we introduce GeoCalib, a deep neural network that leverages universal rules of 3D geometry through an optimization process. GeoCalib is trained end-to-end to estimate camera parameters and learns to find useful visual cues from the data. Experiments on various benchmarks show that GeoCalib is more robust and more accurate than existing classical and learned approaches. Its internal optimization estimates uncertainties, which help flag failure cases and benefit downstream applications like visual localization. The code and trained models are publicly available at https://github.com/cvg/GeoCalib.

We introduce PyTorch Geometric, a library for deep learning on irregularly structured input data such as graphs, point clouds and manifolds, built upon PyTorch. In addition to general graph data structures and processing methods, it contains a variety of recently published methods from the domains of relational learning and 3D data processing. PyTorch Geometric achieves high data throughput by leveraging sparse GPU acceleration, by providing dedicated CUDA kernels and by introducing efficient mini-batch handling for input examples of different size. In this work, we present the library in detail and perform a comprehensive comparative study of the implemented methods in homogeneous evaluation scenarios.

We develop data-driven methods incorporating geometric and topological information to learn parsimonious representations of nonlinear dynamics from observations. The approaches learn nonlinear state-space models of the dynamics for general manifold latent spaces using training strategies related to Variational Autoencoders (VAEs). Our methods are referred to as Geometric Dynamic (GD) Variational Autoencoders (GD-VAEs). We learn encoders and decoders for the system states and evolution based on deep neural network architectures that include general Multilayer Perceptrons (MLPs), Convolutional Neural Networks (CNNs), and other architectures. Motivated by problems arising in parameterized PDEs and physics, we investigate the performance of our methods on tasks for learning reduced dimensional representations of the nonlinear Burgers Equations, Constrained Mechanical Systems, and spatial fields of Reaction-Diffusion Systems. GD-VAEs provide methods that can be used to obtain representations in manifold latent spaces for diverse learning tasks involving dynamics.

We introduce Geometric Neural Operators (GNPs) for accounting for geometric contributions in data-driven deep learning of operators. We show how GNPs can be used (i) to estimate geometric properties, such as the metric and curvatures, (ii) to approximate Partial Differential Equations (PDEs) on manifolds, (iii) learn solution maps for Laplace-Beltrami (LB) operators, and (iv) to solve Bayesian inverse problems for identifying manifold shapes. The methods allow for handling geometries of general shape including point-cloud representations. The developed GNPs provide approaches for incorporating the roles of geometry in data-driven learning of operators.

Generalist robot policies must follow user instructions while reasoning about how objects, cameras, and robot actions interact in the 3D physical world. Recent vision-language-action models (VLAs) and video world-action models (WAMs) inherit strong semantic or temporal priors from large-scale foundation models, but they still operate primarily on 2D image frames or 2D-derived latent spaces, leaving implicit the 3D geometry required for contact-rich manipulation. We propose the Geometric Action Model (GAM), a language-conditioned manipulation policy that directly repurposes a pretrained geometric foundation model (GFM) as a shared substrate for perception, temporal prediction, and action decoding. GAM splits the GFM at an intermediate layer: the shallow layers serve as an observation encoder, and a causal future predictor inserted at the split layer forecasts future latent tokens conditioned on language, proprioception, and action history. The predicted future tokens are then routed through the remaining GFM blocks for feature propagation and decoding, allowing a single backbone to produce both future geometry and actions. This design equips the GFM with language-conditioned temporal world modeling through minimal architectural modification while preserving its rich geometric priors. Across a broad suite of simulation and real-robot manipulation benchmarks, GAM is more accurate, more robust, faster, and lighter than current foundation-model-scale baselines.

Shape assembly, the process of combining parts into a complete whole, is a crucial robotic skill with broad real-world applications. Among various assembly tasks, geometric assembly--where broken parts are reassembled into their original form (e.g., reconstructing a shattered bowl)--is particularly challenging. This requires the robot to recognize geometric cues for grasping, assembly, and subsequent bimanual collaborative manipulation on varied fragments. In this paper, we exploit the geometric generalization of point-level affordance, learning affordance aware of bimanual collaboration in geometric assembly with long-horizon action sequences. To address the evaluation ambiguity caused by geometry diversity of broken parts, we introduce a real-world benchmark featuring geometric variety and global reproducibility. Extensive experiments demonstrate the superiority of our approach over both previous affordance-based and imitation-based methods. Project page: https://sites.google.com/view/biassembly/.

Geometric analysis fundamentally distinguishes between extrinsic and intrinsic perspectives. The dominant paradigm in current 3D representation learning relies on either extrinsic spatial structures or high-level semantics, struggling to capture the essence of shape identity and underlying manifold topology. To bridge this gap, we introduce a novel 3D representation learning paradigm, namely PRISM, for Pre-training, which learns isometric embeddings by Recovering the Intrinsic Surface geodesic Metric. PRISM incorporates a topology-enforcing objective that explicitly constrains the structure of latent space, alongside a specialized two-stage training recipe mitigating sample imbalance inherent in the distribution of geodesic distances. Experiments demonstrate that our approach shows satisfactory accuracy, robustness, and high efficiency in geodesic distance prediction and achieves superior performance across diverse downstream tasks, including shape recognition, surface parameterization, and non-rigid correspondence. The code will be publicly available at https://github.com/AidenZhao/PRISM.

When zooming between dual cameras on smartphones, prominent discontinuities in both geometric perspective and color rendition inevitably degrade the visual experience of users. To address this issue, we introduce the dual-camera smooth zoom (DCSZ) task, aiming to synthesize continuous intermediate frames for fluid zooming. However, directly applying existing video frame interpolation (VFI) models to this task is severely hindered by substantial domain gaps and the practical intractability of capturing real-world continuous ground-truth. To this end, we propose a novel data synthesis framework to generate high-fidelity DCSZ training sequences, named ZoomGS v2. It explicitly decouples geometric and color transformations from 3D scene representation modeling, thereby enabling extensive data generation from easily accessible single-camera 3D scenes. Furthermore, we develop ZoomFI, a specialized VFI network tailored to the DCSZ task, which leverages complementary bidirectional warping for comprehensive dual-camera information fusion. Extensive experiments on test datasets captured using two mobile phones demonstrate that training with the constructed DCSZ data significantly improves the performance of various VFI models. Moreover, ZoomFI achieves state-of-the-art results in both quantitative metrics and visual quality. The datasets, code, and pretrained models are available at https://hitjunjief.github.io/DCSZ-v2

Benefiting from contrastively trained visual encoders on large-scale natural scene images, Large Multimodal Models (LMMs) have achieved remarkable performance across various visual perception tasks. However, the inherent limitations of contrastive learning upon summarized descriptions fundamentally restrict the capabilities of models in meticulous reasoning, particularly in crucial scenarios of geometric problem-solving. To enhance geometric understanding, we propose a novel hard negative contrastive learning framework for the vision encoder, which combines image-based contrastive learning using generation-based hard negatives created by perturbing diagram generation code, and text-based contrastive learning using rule-based negatives derived from modified geometric descriptions and retrieval-based negatives selected based on caption similarity. We train CLIP using our strong negative learning method, namely MMCLIP (Multimodal Math CLIP), and subsequently train an LMM for geometric problem-solving. Experiments show that our trained model, MMGeoLM, significantly outperforms other open-source models on three geometric reasoning benchmarks. Even with a size of 7B, it can rival powerful closed-source models like GPT-4o. We further study the impact of different negative sample construction methods and the number of negative samples on the geometric reasoning performance of LMM, yielding fruitful conclusions. The code and dataset are available at https://github.com/THU-KEG/MMGeoLM.

Affine correspondences have received significant attention due to their benefits in tasks like image matching and pose estimation. Existing methods for extracting affine correspondences still have many limitations in terms of performance; thus, exploring a new paradigm is crucial. In this paper, we present a new pipeline designed for extracting accurate affine correspondences by integrating dense matching and geometric constraints. Specifically, a novel extraction framework is introduced, with the aid of dense matching and a novel keypoint scale and orientation estimator. For this purpose, we propose loss functions based on geometric constraints, which can effectively improve accuracy by supervising neural networks to learn feature geometry. The experimental show that the accuracy and robustness of our method outperform the existing ones in image matching tasks. To further demonstrate the effectiveness of the proposed method, we applied it to relative pose estimation. Affine correspondences extracted by our method lead to more accurate poses than the baselines on a range of real-world datasets. The code is available at https://github.com/stilcrad/DenseAffine.

We introduce Talk2Move, a reinforcement learning (RL) based diffusion framework for text-instructed spatial transformation of objects within scenes. Spatially manipulating objects in a scene through natural language poses a challenge for multimodal generation systems. While existing text-based manipulation methods can adjust appearance or style, they struggle to perform object-level geometric transformations-such as translating, rotating, or resizing objects-due to scarce paired supervision and pixel-level optimization limits. Talk2Move employs Group Relative Policy Optimization (GRPO) to explore geometric actions through diverse rollouts generated from input images and lightweight textual variations, removing the need for costly paired data. A spatial reward guided model aligns geometric transformations with linguistic description, while off-policy step evaluation and active step sampling improve learning efficiency by focusing on informative transformation stages. Furthermore, we design object-centric spatial rewards that evaluate displacement, rotation, and scaling behaviors directly, enabling interpretable and coherent transformations. Experiments on curated benchmarks demonstrate that Talk2Move achieves precise, consistent, and semantically faithful object transformations, outperforming existing text-guided editing approaches in both spatial accuracy and scene coherence.

Deep Neural Networks are highly susceptible to shortcut learning, frequently memorizing low-dimensional spurious correlations instead of underlying causal mechanisms. This phenomenon not only degrades out-of-distribution robustness but also induces severe demographic biases in sensitive applications. In this paper, we propose a geometric a priori methodology to mitigate shortcut learning. By deploying a zero-hidden-layer (N=1) Topological Auditor, we mathematically isolate features that monopolize the gradient without human intervention. We empirically demonstrate a Capacity Phase Transition: once linear shortcuts are pruned, networks are forced to utilize higher geometric capacity (N geq 16) to curve the decision boundary and learn ethical representations. Our approach outperforms L1 Regularization -- which collapses into demographic bias -- and operates at a fraction of the computational cost of post-hoc methods like Just Train Twice (JTT), successfully reducing counterfactual gender vulnerability from 21.18\% to 7.66\%.

Functional magnetic resonance imaging techniques benefit from echo-planar imaging's fast image acquisition but are susceptible to inhomogeneities in the main magnetic field, resulting in geometric distortion and signal loss artifacts in the images. Traditional methods leverage a field map or voxel displacement map for distortion correction. However, voxel displacement map estimation requires additional sequence acquisitions, and the accuracy of the estimation influences correction performance. This work implements a novel approach called GDCNet, which estimates a geometric distortion map by non-linear registration to T1-weighted anatomical images and applies it for distortion correction. GDCNet demonstrated fast distortion correction of functional images in retrospectively and prospectively acquired datasets. Among the compared models, the 2D self-supervised configuration resulted in a statistically significant improvement to normalized mutual information between distortion-corrected functional and T1-weighted images compared to the benchmark methods FUGUE and TOPUP. Furthermore, GDCNet models achieved processing speeds 14 times faster than TOPUP in the prospective dataset.

We introduce VEGA, an approach for training navigation VisionLanguage-Action (VLA) models from unlabeled egocentric navigation videos. Internet-scale egocentric videos provide a scalable source of navigation-relevant visual observations, capturing cluttered scenes, close-range obstacles, and natural human motion through real-world spaces. However, these videos are not directly usable for policy learning because they do not provide obstacle-aware trajectories conditioned on explicit navigation goals in the robot's coordinate frame. VEGA addresses this gap by reconstructing local scene geometry from monocular video, sampling navigation goals (represented as text, image, or spatial waypoints) and generating obstacle-aware trajectories using the constructed geometry. The resulting trajectory distribution is then used to train a flow-matching VLA navigation policy. By using geometry exclusively during training, VEGA distills obstacle-aware planning directly into a vision-based policy. Furthermore, we introduce VEGA-Bench, a benchmark containing 250k scenes and approximately 5 million navigation goals paired with scene geometry, designed to evaluate goal progress, collision avoidance, and obstacle clearance of VLAs. Our evaluation shows that VEGA achieves competitive goal progress while reducing collisions by 33.0% and improving obstacle clearance by 17.9% over the strongest baseline on VEGABench, while improving success by at least 150.0%, reducing collisions by at least 66.7%, and improving obstacle clearance by at least 60.0% in real-world trials. Ultimately, we demonstrate that video-derived geometric supervision provides a scalable and effective signal for training obstacle-aware navigation VLAs. The code and benchmark will be released at the time of publication.

We study how trajectory-shape diversity in demonstrations affects imitation learning (IL) performance across models, tasks, and data scales. We introduce Geometric Entropy (H_G), a task-agnostic metric that quantifies the intrinsic diversity of transit trajectories after normalizing away extrinsic variation, such as goal pose and workspace scale, via target-frame alignment. Across multiple IL architectures and both simulated and real-robot contact-rich manipulation tasks, we observe a consistent inverted-U relationship between success and H_G: increasing geometric diversity improves robustness in low-diversity regimes but degrades performance once diversity induces strategy ambiguity. Moreover, the optimal entropy shifts toward lower values as task mastery increases through more data, easier tasks, or stronger priors, and for a pretrained vision-language-action model the trend becomes effectively monotonic decreasing. Practically, H_G enables fast pre-training auditing of demonstration datasets and offers a simple guideline for calibrating demonstrations toward the learnable regime.

World-action models (WAMs) predict scene changes to guide robot actions, yet their RGB futures entangle state changes with appearance and lack 3D geometry. Recent WAMs add geometry as a training signal or a control interface, or build native 3D models, but decide what geometry to model rather than how the action should read it. Intuitively, control-relevant cues lie at embodiment--object interactions and their transitions, yet probing existing WAMs reveals that their action attention underweights these interactions in favor of the rest of the embodiment. To focus the action on these interactions, we present Gnomon4D, a WAM that jointly generates future video, robot actions, and GeoDelta, a video-aligned 4D geometric motion field recording the image-plane displacement and depth change of embodiment and object points. With the embodiment and objects in one field, their relative motion, which marks an interaction, can be read off. Guided by this motion and the task, Interaction-Salient Attention (InSAttn) selects where the action reads, and Transition-Anchored Aggregation (TransAgg) selects when, around grasp and release. As GeoDelta encodes surface motion rather than embodiment-specific controls, it serves as a human--robot quasi-action that links action-free pretraining on EgoManip-5000H, which we curate, to robot mid-training on Robot-30H. Gnomon4D leads on RoboTwin~2.0, VLABench, and RoboDojo, and outperforms real-robot baselines, especially under visual shifts. Project page: https://droliven.github.io/Gnomon4D.

Vision-language models (VLMs) often struggle with geometric reasoning due to their limited perception of fundamental diagram elements. To tackle this challenge, we introduce GeoPerceive, a benchmark comprising diagram instances paired with domain-specific language (DSL) representations, along with an efficient automatic data generation pipeline. This design enables the isolated evaluation of geometric perception independently from reasoning. To exploit the data provided by GeoPerceive for enhancing the geometric perception capabilities of VLMs, we propose GeoDPO, a translator-guided reinforcement learning (RL) framework. GeoDPO employs an NL-to-DSL translator, which is trained on synthetic pairs generated by the data engine of GeoPerceive, to bridge natural language and DSL. This translator facilitates the computation of fine-grained, DSL-level scores, which serve as reward signals in reinforcement learning. We assess GeoDPO on both in-domain and out-of-domain datasets, spanning tasks in geometric perception as well as downstream reasoning. Experimental results demonstrate that, while supervised fine-tuning (SFT) offers only marginal improvements and may even impair performance in out-of-domain scenarios, GeoDPO achieves substantial gains: +26.5% on in-domain data, +8.0% on out-of-domain data, and +39.0% on downstream reasoning tasks. These findings underscore the superior performance and generalization ability of GeoDPO over SFT. All codes are released at https://github.com/Longin-Yu/GeoPerceive to ensure reproducibility.