Daily Papers

Recent 3D world models increasingly use features from geometry foundation models (GFMs) as predictive world states, benefiting from their rich geometric priors. However, the view dependence of these features entangles scene dynamics with camera-induced variation, making the resulting state evolution harder to predict. We propose GeoState, a view-decoupled geometry representation for predictive 3D world modeling that separates scene-state evolution from camera-dependent observation. Specifically, a scene tokenizer aggregates multi-view GFM features into a view-decoupled scene state, which a geometry decoder queries using camera rays. An autoregressive world model then predicts future scene states, while camera motion is modeled separately. The predicted states can therefore be decoded under either predicted or externally specified cameras, while retaining a single context-derived scale throughout the rollout without per-frame scale alignment. Experiments show that GeoState outperforms prior 3D world models on Waymo, supports geometrically consistent cross-view querying, and generalizes zero-shot to KITTI and nuScenes. Project page: https://hongchang-liu.github.io/GeoState-project-page/

Point cloud upsampling is essential for 3D applications but remains challenging due to sparse, under-constrained observations. Multi-view images can resolve geometric ambiguity, but existing multi-modal methods do not fully exploit visual cues. We propose ControlFlow3D, a framework that distills knowledge from pretrained foundation models into a lightweight generation network during training, enabling strong performance without multi-view images available at test time. Specifically, our method performs conditional flow matching in a structured latent space defined by a frozen 3D encoder. Multi-view image tokens and 3D latent tokens are jointly clustered into shared prototypes via unsupervised soft KMeans; the resulting prototypes guide the flow trajectory through velocity control and are aligned by a consistency loss that progressively transfers image knowledge into the base network. A Mamba-based geometry decoder then maps latent features to dense coordinates with linear complexity. We also construct ShapeNetPU, a 35K-object multi-modal benchmark that is 31 times larger than prior datasets. Extensive experiments across multiple benchmarks demonstrate the effectiveness of our method and its strong generalization capability.

We present a novel meta-learning approach for 6D pose estimation on unknown objects. In contrast to ``instance-level" and ``category-level" pose estimation methods, our algorithm learns object representation in a category-agnostic way, which endows it with strong generalization capabilities across object categories. Specifically, we employ a neural process-based meta-learning approach to train an encoder to capture texture and geometry of an object in a latent representation, based on very few RGB-D images and ground-truth keypoints. The latent representation is then used by a simultaneously meta-trained decoder to predict the 6D pose of the object in new images. Furthermore, we propose a novel geometry-aware decoder for the keypoint prediction using a Graph Neural Network (GNN), which explicitly takes geometric constraints specific to each object into consideration. To evaluate our algorithm, extensive experiments are conducted on the \linemod dataset, and on our new fully-annotated synthetic datasets generated from Multiple Categories in Multiple Scenes (MCMS). Experimental results demonstrate that our model performs well on unseen objects with very different shapes and appearances. Remarkably, our model also shows robust performance on occluded scenes although trained fully on data without occlusion. To our knowledge, this is the first work exploring cross-category level 6D pose estimation.

Joint-embedding predictive architectures (JEPAs) propose that a model should learn more useful abstractions when trained to predict latent representations rather than observed outputs. For autoregressive language-model fine-tuning the principle entails a stricter requirement: the induced hidden-state geometry must reach the language-model head and improve the decoded task metric. We test that requirement under a fixed Llama-3.2-1B-Instruct LoRA harness on natural-language-to-regex generation, comparing twenty-two training-time auxiliaries across trajectory-shape regularisation, distributional constraints, predictor/target asymmetry, Fisher-metric Jacobi residuals, and a decoder-visible JEPA objective constructed to lie in cross-entropy's positive cone. The empirical answer is a structured null: several auxiliaries clear single-cell paired α= 0.10 without correction (T3-Local at Δ= +2.53~pp, p = 0.003 being the strongest), but none survives Bonferroni or Holm--Bonferroni at the relevant family-wise threshold, even though many change curvature, anisotropy, variance, and gradient direction. Decoder-visible JEPA yields the first positive auxiliary--cross-entropy gradient cosine in the study, yet exact match remains inside seed noise; a full-fine-tuning replication of the same auxiliary at n = 5 seeds reproduces the null on both benchmarks (TURK: Δ= +0.04~pp, p_{paired} = 0.96; SYNTH: Δ= +0.52~pp, p_{paired} = 0.28), so the null is robust across LoRA and full fine-tuning for the decoder-visible construction. Hidden-state representation work and decoded-task accuracy are therefore weakly coupled in this regime; we accordingly reframe LLM-domain JEPA evaluation as a coupling problem, in which the operative question is under which metrics useful hidden geometry becomes decoder-visible task signal.

Geometry Foundation Models (GFMs) have substantially advanced monocular 3D reconstruction, yet extending this capability to 4D dynamic understanding remains a fundamental challenge. Most existing motion perception methods (e.g., sparse tracking, dense point-wise flow) treat motion as independent point-wise displacements, ignoring the structured nature of physical motion. However, real-world objects usually obey rigid-body kinematics, and points thus usually move collectively, not in isolation. Motion itself possesses geometric structure: physical objects undergo a set of rigid-body transformations governed by SE(3), rather than unstructured point-wise displacements. Building on this insight, we propose SM4RT, a Structured Motion 4D Reconstruction Transformer for end-to-end 3D reconstruction and structured motion perception. SM4RT introduces Structure-of-Motion to represent scene dynamics, where scene motion is decomposed into a compact set of motion bases, each represented as a temporal sequence of 6D twists in SE(3). Dense scene motion is then recovered by sparse, time-shared per-pixel assignment weights over these bases, ensuring points on the same object share a common rigid-body motion trajectory. SM4RT introduces a parallel motion geometry encoder and decoder that jointly infer 3D geometry, world-coordinate motion, and scene kinematic structure in a single forward pass from monocular RGB video. SM4RT achieves strong motion reconstruction performance while preserving the geometric structure of scene motion.

While feed-forward 3D Gaussian Splatting (3DGS) enables efficient 3D reconstruction, achieving high-fidelity rendering remains challenging. Existing pixel-aligned approaches suffer from spatial inflexibility and massive structural redundancy, whereas query-based methods lack 3D priors and entangle geometry with appearance, yielding blurry, pose-dependent results. To overcome these deficiencies, we propose QuerySplat, a feed-forward 3DGS framework driven by geometric priors and explicit appearance decoupling. Specifically, we design a dual-branch query-based decoder: the geometry branch leverages a pretrained Vision Geometric Model for spatial understanding, which intrinsically endows QuerySplat with pose-free modeling capabilities, while the appearance branch recovers high-frequency details through a dedicated pathway separated from geometric attribute regression. Extensive experiments demonstrate that QuerySplat mitigates the blurry rendering issues of early query-based models and consistently outperforms pixel-aligned approaches in rendering fidelity. On the challenging DL3DV benchmark, it achieves state-of-the-art novel view synthesis performance, with average PSNR gains of 2.30 dB and 1.04 dB over the best pose-free and pose-required baselines, respectively. Project Page: https://inspatio.github.io/querysplat.

We introduce a framework for learning latent representations of 4D objects which are descriptive, faithfully capturing object geometry and appearance; compressive, aiding in downstream efficiency; and accessible, requiring minimal input, i.e., an unstructured dynamic point cloud, to construct. Specifically, Velox trains an encoder to compress spatiotemporal color point clouds into a set of dynamic shape tokens. These tokens are supervised using two complementary decoders: a 4D surface decoder, which models the time-varying surface distribution capturing the geometry; and a Gaussian decoder, which maps the tokens to 3D Gaussians, helping learn appearance. To demonstrate the utility of our representation, we evaluate it across three downstream tasks -- video-to-4D generation, 3D tracking, and cloth simulation via image-to-4D generation -- and observe strong performances in all settings.

We present CARTO, a novel approach for reconstructing multiple articulated objects from a single stereo RGB observation. We use implicit object-centric representations and learn a single geometry and articulation decoder for multiple object categories. Despite training on multiple categories, our decoder achieves a comparable reconstruction accuracy to methods that train bespoke decoders separately for each category. Combined with our stereo image encoder we infer the 3D shape, 6D pose, size, joint type, and the joint state of multiple unknown objects in a single forward pass. Our method achieves a 20.4% absolute improvement in mAP 3D IOU50 for novel instances when compared to a two-stage pipeline. Inference time is fast and can run on a NVIDIA TITAN XP GPU at 1 HZ for eight or less objects present. While only trained on simulated data, CARTO transfers to real-world object instances. Code and evaluation data is available at: http://carto.cs.uni-freiburg.de

Despite their proficiency in general tasks, Multi-modal Large Language Models (MLLMs) struggle with automatic Geometry Problem Solving (GPS), which demands understanding diagrams, interpreting symbols, and performing complex reasoning. This limitation arises from their pre-training on natural images and texts, along with the lack of automated verification in the problem-solving process. Besides, current geometric specialists are limited by their task-specific designs, making them less effective for broader geometric problems. To this end, we present GeoX, a multi-modal large model focusing on geometric understanding and reasoning tasks. Given the significant differences between geometric diagram-symbol and natural image-text, we introduce unimodal pre-training to develop a diagram encoder and symbol decoder, enhancing the understanding of geometric images and corpora. Furthermore, we introduce geometry-language alignment, an effective pre-training paradigm that bridges the modality gap between unimodal geometric experts. We propose a Generator-And-Sampler Transformer (GS-Former) to generate discriminative queries and eliminate uninformative representations from unevenly distributed geometric signals. Finally, GeoX benefits from visual instruction tuning, empowering it to take geometric images and questions as input and generate verifiable solutions. Experiments show that GeoX outperforms both generalists and geometric specialists on publicly recognized benchmarks, such as GeoQA, UniGeo, Geometry3K, and PGPS9k.

We introduce GeoDANO, a geometric vision-language model (VLM) with a domain-agnostic vision encoder, for solving plane geometry problems. Although VLMs have been employed for solving geometry problems, their ability to recognize geometric features remains insufficiently analyzed. To address this gap, we propose a benchmark that evaluates the recognition of visual geometric features, including primitives such as dots and lines, and relations such as orthogonality. Our preliminary study shows that vision encoders often used in general-purpose VLMs, e.g., OpenCLIP, fail to detect these features and struggle to generalize across domains. We develop GeoCLIP, a CLIP based model trained on synthetic geometric diagram-caption pairs to overcome the limitation. Benchmark results show that GeoCLIP outperforms existing vision encoders in recognizing geometric features. We then propose our VLM, GeoDANO, which augments GeoCLIP with a domain adaptation strategy for unseen diagram styles. GeoDANO outperforms specialized methods for plane geometry problems and GPT-4o on MathVerse.

In this work, instead of directly predicting the pixel-level segmentation masks, the problem of referring image segmentation is formulated as sequential polygon generation, and the predicted polygons can be later converted into segmentation masks. This is enabled by a new sequence-to-sequence framework, Polygon Transformer (PolyFormer), which takes a sequence of image patches and text query tokens as input, and outputs a sequence of polygon vertices autoregressively. For more accurate geometric localization, we propose a regression-based decoder, which predicts the precise floating-point coordinates directly, without any coordinate quantization error. In the experiments, PolyFormer outperforms the prior art by a clear margin, e.g., 5.40% and 4.52% absolute improvements on the challenging RefCOCO+ and RefCOCOg datasets. It also shows strong generalization ability when evaluated on the referring video segmentation task without fine-tuning, e.g., achieving competitive 61.5% J&F on the Ref-DAVIS17 dataset.

Interaction understanding in 3D scenes requires a joint description of movable parts, their motion, and the regions through which they can be operated. We present Segment-Snap, which connects these outputs through the physical relationship between parts and handles. Learned predictors identify broad part surfaces and small handles. A geometric decoder uses planar and upright priors to constrain motion, then selects hinge lines using predicted handle locations, without training a motion regressor. Conversely, a joint part-and-handle predictor supplies additional handle candidates, whose motion classes are refined using containing parts. Each information transfer is applied once, without iterative feedback. On Articulate3D validation, handle guidance raises motion-gated AP from 13.74% to 40.98% at fixed masks and axes. Additional handle candidates raise handle AP from 24.63% to 29.65%; part-based class correction adds 0.98 points, and full context reaches 30.99%. Repeated training, learned-decoder controls and paired visualizations establish the benefits and limitations of combining geometric and semantic evidence for interaction understanding.

Recent advances in Neural Fields have enabled powerful, discretization-invariant methods for learning neural operators that approximate solutions of Partial Differential Equations (PDEs) on general geometries. Building on these developments, we introduce enf2enf, an encoder--decoder methodology for predicting steady-state Partial Differential Equations with non-parameterized geometric variability, based on recently proposed Equivariant Neural Field architectures. In enf2enf, input geometries are encoded into latent point cloud embeddings that inherently preserve geometric grounding and capture local phenomena. The resulting representations are then combined with global parameters and directly decoded into continuous output fields, thus efficiently modeling the coupling between geometry and physics. By leveraging the inductive biases of locality and translation invariance, our approach is able to capture fine-scale physical features as well as complex shape variations, thereby enhancing generalization and physical compliance. Extensive experiments on a high-fidelity aerodynamic dataset, a hyper-elastic material benchmark, and multi-element airfoil geometries, demonstrate that the proposed model achieves superior or competitive performance compared to state-of-the-art graph based, operator learning, and neural field methods. Notably, our method supports real time inference and zero-shot super-resolution, enabling efficient training on low-resolution meshes while maintaining high accuracy on full-scale discretizations.

Plane geometry problem solving (PGPS) has recently gained significant attention as a benchmark to assess the multi-modal reasoning capabilities of large vision-language models. Despite the growing interest in PGPS, the research community still lacks a comprehensive overview that systematically synthesizes recent work in PGPS. To fill this gap, we present a survey of existing PGPS studies. We first categorize PGPS methods into an encoder-decoder framework and summarize the corresponding output formats used by their encoders and decoders. Subsequently, we classify and analyze these encoders and decoders according to their architectural designs. Finally, we outline major challenges and promising directions for future research. In particular, we discuss the hallucination issues arising during the encoding phase within encoder-decoder architectures, as well as the problem of data leakage in current PGPS benchmarks.

The very challenging task of learning solution operators of PDEs on arbitrary domains accurately and efficiently is of vital importance to engineering and industrial simulations. Despite the existence of many operator learning algorithms to approximate such PDEs, we find that accurate models are not necessarily computationally efficient and vice versa. We address this issue by proposing a geometry aware operator transformer (GAOT) for learning PDEs on arbitrary domains. GAOT combines novel multiscale attentional graph neural operator encoders and decoders, together with geometry embeddings and (vision) transformer processors to accurately map information about the domain and the inputs into a robust approximation of the PDE solution. Multiple innovations in the implementation of GAOT also ensure computational efficiency and scalability. We demonstrate this significant gain in both accuracy and efficiency of GAOT over several baselines on a large number of learning tasks from a diverse set of PDEs, including achieving state of the art performance on a large scale three-dimensional industrial CFD dataset.

Discrete motion tokenization has recently enabled Large Language Models (LLMs) to serve as versatile backbones for motion understanding and motion-language reasoning. However, existing pipelines typically decouple motion quantization from semantic embedding learning, linking them solely via token IDs. This approach fails to effectively align the intrinsic geometry of the motion space with the embedding space, thereby hindering the LLM's capacity for nuanced motion reasoning. We argue that alignment is most effective when both modalities share a unified geometric basis. Therefore, instead of forcing the LLM to reconstruct the complex geometry among motion tokens from scratch, we present a novel framework that explicitly enforces orthogonality on both the motion codebook and the LLM embedding space, ensuring that their relational structures naturally mirror each other. Specifically, we employ a decoder-only quantizer with Gumbel-Softmax for differentiable training and balanced codebook usage. To bridge the modalities, we use a sparse projection that maps motion codes into the LLM embedding space while preserving orthogonality. Finally, a two-stage orthonormal regularization schedule enforces soft constraints during tokenizer training and LLM fine-tuning to maintain geometric alignment without hindering semantic adaptation. Extensive experiments on HumanML3D demonstrate that our framework achieves a 20% performance improvement over current state-of-the-art methods, validating that a unified geometric basis effectively empowers the LLM for nuanced motion reasoning.

Recent work has shown that neural networks can perform 3D tasks such as Novel View Synthesis (NVS) without explicit 3D reconstruction. Even so, we argue that strong 3D inductive biases are still helpful in the design of such networks. We show this point by introducing LagerNVS, an encoder-decoder neural network for NVS that builds on `3D-aware' latent features. The encoder is initialized from a 3D reconstruction network pre-trained using explicit 3D supervision. This is paired with a lightweight decoder, and trained end-to-end with photometric losses. LagerNVS achieves state-of-the-art deterministic feed-forward Novel View Synthesis (including 31.4 PSNR on Re10k), with and without known cameras, renders in real time, generalizes to in-the-wild data, and can be paired with a diffusion decoder for generative extrapolation.

Multi-view 3D reconstruction has achieved remarkable progress with the advent of feed-forward 3D reconstruction models. However, these models are typically trained and evaluated under ideal, degradation-free imaging conditions, whereas real-world observations often contain degradations that differ significantly from such settings. Improving robustness for multi-view 3D reconstruction under degraded conditions therefore remains an important challenge. We present Geometry-Aware Representation Denoising (GARD), a novel framework that performs diffusion-based multi-view restoration directly in the feature space of a feed-forward 3D reconstruction model. This design exploits the geometry-aware feature representations of the 3D reconstructor to effectively recover accurate scene geometry. Furthermore, by employing an additional RGB image decoder, the refined representations can also be used to restore high-quality RGB images, thereby enabling the simultaneous recovery of 3D scene geometry and high-quality imagery. Comprehensive experiments on the Depth Anything 3 (DA3) benchmark demonstrate the effectiveness of the proposed GARD framework.

Diffusion-based models enable monocular geometry estimation, yet their pixel-space precision is limited by a shared, under-studied error source: VAE reconstruction degradation. The 8x spatial compression in the VAE encoder-decoder degrades surface normals at object boundaries; even encoding and decoding ground-truth normals introduces 1.3--8.5° of mean angular error (MAE), with edge MAE reaching 2.8x the global MAE. We present TransNormal-2, a FLUX.2-based rectified-flow framework with single-step deterministic inference that addresses this degradation on both sides of the VAE decoder: in how latent predictions are supervised during training, and in how decoded normals are corrected at inference. First, geometry-aware pixel-space losses, including inverse rendering self-consistency, von~Mises-Fisher angular loss, and wavelet edge-aware regularization, complement latent MSE by enforcing spherical normal geometry and diffuse image-formation cues after VAE decoding. Second, a lightweight Geometric Refinement Module (GRM) applies an RGB-guided residual correction to reduce boundary-localized decoding errors without freely rewriting the coarse prediction. On general-scene benchmarks, TransNormal-2 matches or exceeds MoGe-2 on all eight reported metrics while using only 1.4% as many task-specific normal annotations. The gains are clearest for transparent objects, reducing MAE by 4.2° on ClearGrasp and 3.1° on ClearPose over the strongest prior baselines. Code will be released at https://longxiang-ai.github.io/TransNormal-2.

Decoder hidden states from different layers of large language models (LLMs) often encode complementary information. However, existing speculative decoding methods typically rely on a small set of target-model layers chosen using fixed, position-based heuristics. Selecting an optimal subset of layers is inherently a combinatorial problem, making exhaustive evaluation infeasible even for moderately deep models. In this work, we present a principled framework for multi-layer hidden-state selection for speculative decoding based on two complementary criteria: informativeness, measured through matrix-entropy analysis, and diversity, quantified using centered kernel alignment (CKA). We investigate layer-wise Shannon entropy, aggregate entropy under hidden-state concatenation, and pairwise CKA dynamics across depth, revealing distinct geometric and informational compartments within decoder representations. Motivated by these findings, we propose entropy and CKA-based layer selection algorithms that favor hidden-state sets that are both highly informative and minimally redundant. As part of this framework, we introduce token-entropy filtering, which removes hidden states associated with low predictive-entropy tokens, thereby improving the quality of layer-selection statistics. In EAGLE3-style speculative decoding, our selected sets outperform fixed baseline choices, including the default [2, N//2, N-3] rule, while providing a clearer explanation of why some defaults can remain competitive. Our results provide a practical recipe for multi-layer hidden-state selection for speculative decoding, insights into the geometry and information in LLM decoder states, and a foundation for extending similar analysis to related applications.

Chunked vision-language-action (VLA) policies predict multi-step robot controls, conditioning each update on the current visual observation alone. Yet robot actions cause contact, occlusion, and object motion, and the geometry that later decisions depend on can change before the next visual update arrives. Spatial VLAs improve current-frame geometry. Temporal VLAs aggregate past frames. Neither maintains an action-updated scene prior across chunks. We argue for a persistent action-updated scene state across control calls, and introduce EvoScene-VLA. Its recurrent scene prefix carries a geometry-aware scene state across chunks. At each vision-language model (VLM) call, the VLM combines scene information from the current observation with the action-updated prior from the previous chunk; the action decoder outputs both the next action chunk and a compact scene update. This update becomes the next prior, which the VLM corrects against the new observation when the next call arrives. Each control call therefore starts from a scene prior that reflects both recent actions and fresh visual evidence. During training, Scene Predictor supplies future scene-token targets, and Geometric Anchor aligns scene slots with frozen depth and 3D teachers. We discard both modules at deployment. On 31 RoboTwin tasks, EvoScene-VLA raises average success from 87.2% to 89.1% in fixed evaluation and from 86.1% to 88.5% in randomized evaluation. On the Galaxea R1-Lite real robot, EvoScene-VLA outperforms all baselines.

We present a novel geometric perspective on the latent space of diffusion models. We first show that the standard pullback approach, utilizing the deterministic probability flow ODE decoder, is fundamentally flawed. It provably forces geodesics to decode as straight segments in data space, effectively ignoring any intrinsic data geometry beyond the ambient Euclidean space. Complementing this view, diffusion also admits a stochastic decoder via the reverse SDE, which enables an information geometric treatment with the Fisher-Rao metric. However, a choice of x_T as the latent representation collapses this metric due to memorylessness. We address this by introducing a latent spacetime z=(x_t,t) that indexes the family of denoising distributions p(x_0 | x_t) across all noise scales, yielding a nontrivial geometric structure. We prove these distributions form an exponential family and derive simulation-free estimators for curve lengths, enabling efficient geodesic computation. The resulting structure induces a principled Diffusion Edit Distance, where geodesics trace minimal sequences of noise and denoise edits between data. We also demonstrate benefits for transition path sampling in molecular systems, including constrained variants such as low-variance transitions and region avoidance. Code is available at: https://github.com/rafalkarczewski/spacetime-geometry.

Perspective projection has been extensively utilized in monocular 3D object detection methods. It introduces geometric priors from 2D bounding boxes and 3D object dimensions to reduce the uncertainty of depth estimation. However, due to depth errors originating from the object's visual surface, the height of the bounding box often fails to represent the actual projected central height, which undermines the effectiveness of geometric depth. Direct prediction for the projected height unavoidably results in a loss of 2D priors, while multi-depth prediction with complex branches does not fully leverage geometric depth. This paper presents a Transformer-based monocular 3D object detection method called MonoDGP, which adopts perspective-invariant geometry errors to modify the projection formula. We also try to systematically discuss and explain the mechanisms and efficacy behind geometry errors, which serve as a simple but effective alternative to multi-depth prediction. Additionally, MonoDGP decouples the depth-guided decoder and constructs a 2D decoder only dependent on visual features, providing 2D priors and initializing object queries without the disturbance of 3D detection. To further optimize and fine-tune input tokens of the transformer decoder, we also introduce a Region Segment Head (RSH) that generates enhanced features and segment embeddings. Our monocular method demonstrates state-of-the-art performance on the KITTI benchmark without extra data. Code is available at https://github.com/PuFanqi23/MonoDGP.

We propose NeRF-VAE, a 3D scene generative model that incorporates geometric structure via NeRF and differentiable volume rendering. In contrast to NeRF, our model takes into account shared structure across scenes, and is able to infer the structure of a novel scene -- without the need to re-train -- using amortized inference. NeRF-VAE's explicit 3D rendering process further contrasts previous generative models with convolution-based rendering which lacks geometric structure. Our model is a VAE that learns a distribution over radiance fields by conditioning them on a latent scene representation. We show that, once trained, NeRF-VAE is able to infer and render geometrically-consistent scenes from previously unseen 3D environments using very few input images. We further demonstrate that NeRF-VAE generalizes well to out-of-distribution cameras, while convolutional models do not. Finally, we introduce and study an attention-based conditioning mechanism of NeRF-VAE's decoder, which improves model performance.

Riemannian representation learning typically relies on approximating densities on chosen manifolds. This involves optimizing difficult objectives, potentially harming models. To completely circumvent this issue, we introduce the Riemannian generative decoder which finds manifold-valued maximum likelihood latents with a Riemannian optimizer while training a decoder network. By discarding the encoder, we vastly simplify the manifold constraint compared to current approaches which often only handle few specific manifolds. We validate our approach on three case studies -- a synthetic branching diffusion process, human migrations inferred from mitochondrial DNA, and cells undergoing a cell division cycle -- each showing that learned representations respect the prescribed geometry and capture intrinsic non-Euclidean structure. Our method requires only a decoder, is compatible with existing architectures, and yields interpretable latent spaces aligned with data geometry.

Recent advances in dense 3D reconstruction have led to significant progress, yet achieving accurate unified geometric prediction remains a major challenge. Most existing methods are limited to predicting a single geometry quantity from input images. However, geometric quantities such as depth, surface normals, and point maps are inherently correlated, and estimating them in isolation often fails to ensure consistency, thereby limiting both accuracy and practical applicability. This motivates us to explore a unified framework that explicitly models the structural coupling among different geometric properties to enable joint regression. In this paper, we present Dens3R, a 3D foundation model designed for joint geometric dense prediction and adaptable to a wide range of downstream tasks. Dens3R adopts a two-stage training framework to progressively build a pointmap representation that is both generalizable and intrinsically invariant. Specifically, we design a lightweight shared encoder-decoder backbone and introduce position-interpolated rotary positional encoding to maintain expressive power while enhancing robustness to high-resolution inputs. By integrating image-pair matching features with intrinsic invariance modeling, Dens3R accurately regresses multiple geometric quantities such as surface normals and depth, achieving consistent geometry perception from single-view to multi-view inputs. Additionally, we propose a post-processing pipeline that supports geometrically consistent multi-view inference. Extensive experiments demonstrate the superior performance of Dens3R across various dense 3D prediction tasks and highlight its potential for broader applications.

Recent feedforward 3D reconstruction methods predict point maps and estimate global 3D geometry remarkably well. However, their predictions still exhibit inaccurate local surface geometry, which is clearly visible qualitatively but only weakly reflected in common metrics. To make these errors more explicit in evaluation, we introduce a point map normal metric that evaluates the local surface orientation induced by neighboring 3D predictions. To reduce these errors, we propose two complementary components: a point gradient matching loss that supervises depth-normalized 3D finite differences, and a Neighborhood Attention Decoder (NAD) that progressively upsamples features and uses Neighborhood Attention for local feature mixing. Across eight zero-shot monocular geometry benchmarks, our model, SurGe, achieves the best average rank for global point map AbsRel and consistently improves local point map and point map normal evaluations.

Reconstructing coherent 3D geometry and appearance from unposed multi-view images is a fundamental yet challenging problem in computer vision. Most existing visual geometry foundation models predict explicit geometry by regressing pixel-aligned pointmaps, often suffering from redundancy and limited geometric continuity. We propose IVGT, an Implicit Visual Geometry Transformer that implicitly models continuous and coherent geometry from pose-free multi-view images. This formulation learns a continuous neural scene representation in a canonical coordinate system and supports continuous spatial queries at any 3D positions, retrieving local features to predict signed distance (SDF) values and colors using lightweight decoders. It allows direct extraction of continuous and coherent surface geometry, enabling rendering of RGB images, depth maps, and surface normal maps from arbitrary viewpoints. We train IVGT via multi-dataset joint optimization with 2D supervision and 3D geometric regularization. IVGT demonstrates generalization across scenes and achieves strong performance on various tasks, including mesh and point cloud reconstruction, novel view synthesis, depth and surface normal estimation, and camera pose estimation.

We introduce MeshGPT, a new approach for generating triangle meshes that reflects the compactness typical of artist-created meshes, in contrast to dense triangle meshes extracted by iso-surfacing methods from neural fields. Inspired by recent advances in powerful large language models, we adopt a sequence-based approach to autoregressively generate triangle meshes as sequences of triangles. We first learn a vocabulary of latent quantized embeddings, using graph convolutions, which inform these embeddings of the local mesh geometry and topology. These embeddings are sequenced and decoded into triangles by a decoder, ensuring that they can effectively reconstruct the mesh. A transformer is then trained on this learned vocabulary to predict the index of the next embedding given previous embeddings. Once trained, our model can be autoregressively sampled to generate new triangle meshes, directly generating compact meshes with sharp edges, more closely imitating the efficient triangulation patterns of human-crafted meshes. MeshGPT demonstrates a notable improvement over state of the art mesh generation methods, with a 9% increase in shape coverage and a 30-point enhancement in FID scores across various categories.

Constructing 4D language fields is crucial for embodied AI, augmented/virtual reality, and 4D scene understanding, as they provide enriched semantic representations of dynamic environments and enable open-vocabulary querying in complex scenarios. However, existing approaches to 4D semantic field construction primarily rely on scene-specific Gaussian splatting, which requires per-scene optimization, exhibits limited generalization, and is difficult to scale to real-world applications. To address these limitations, we propose 4DLangVGGT, the first Transformer-based feed-forward unified framework for 4D language grounding, that jointly integrates geometric perception and language alignment within a single architecture. 4DLangVGGT has two key components: the 4D Visual Geometry Transformer, StreamVGGT, which captures spatio-temporal geometric representations of dynamic scenes; and the Semantic Bridging Decoder (SBD), which projects geometry-aware features into a language-aligned semantic space, thereby enhancing semantic interpretability while preserving structural fidelity. Unlike prior methods that depend on costly per-scene optimization, 4DLangVGGT can be jointly trained across multiple dynamic scenes and directly applied during inference, achieving both deployment efficiency and strong generalization. This design significantly improves the practicality of large-scale deployment and establishes a new paradigm for open-vocabulary 4D scene understanding. Experiments on HyperNeRF and Neu3D datasets demonstrate that our approach not only generalizes effectively but also achieves state-of-the-art performance, achieving up to 2% gains under per-scene training and 1% improvements under multi-scene training. Our code released in https://github.com/hustvl/4DLangVGGT

3D foundation models (3DFMs) have recently transformed 3D vision, enabling joint prediction of depths, poses, and point maps directly from images. Yet their ability to reason under extreme, non-overlapping views remains largely unexplored. In this work, we study their internal representations and find that 3DFMs exhibit an emergent understanding of extreme-view geometry, despite never being trained for such conditions. To further enhance these capabilities, we introduce a lightweight alignment scheme that refines their internal 3D representation by tuning only a small subset of backbone bias terms, leaving all decoder heads frozen. This targeted adaptation substantially improves relative pose estimation under extreme viewpoints without degrading per-image depth or point quality. Additionally, we contribute MegaUnScene, a new benchmark of Internet scenes unseen by existing 3DFMs, with dedicated test splits for both relative pose estimation and dense 3D reconstruction. All code and data will be released.

Latent flow matching for image generation usually transports Gaussian noise to variational autoencoder latents along linear paths. Both endpoints, however, concentrate in thin spherical shells, and a Euclidean chord leaves those shells even when preprocessing aligns their radii. By decomposing each latent token into radial and angular components, we show through component-swap probes that decoded perceptual and semantic content is carried predominantly by direction, with radius contributing much less. We therefore project data latents onto a fixed token radius, use the radial projection of Gaussian noise as the spherical prior, finetune the decoder with the encoder frozen, and replace linear interpolation with spherical linear interpolation. The resulting geodesic paths stay on the sphere at every timestep, and their velocity targets are purely angular by construction. Under matched training, the method consistently improves class-conditional ImageNet-256 FID across different image tokenizers, leaves the diffusion architecture unchanged, and requires no auxiliary encoder or representation-alignment objective.

Reconstructing articulated objects from sparse images requires recovering complete geometry, movable parts, and motion parameters. Recent methods typically separate geometry reconstruction, part reasoning, and articulation estimation into different stages. This separation can weaken consistency between shape, active parts, and motion, while also incurring substantial inference cost. We introduce Artic-O, an end-to-end, feed-forward framework for articulated object reconstruction via latent geometry learning. Instead of fitting geometry in image or view space, Artic-O maps sparse multi-state observations into a pretrained latent geometry space, where a frozen flow-matching decoder provides a complete-shape prior for recovering visible and occluded structures. To connect geometry with articulation, Artic-O fuses visual tokens, geometry latents, and point-wise decoder features in an image-grounded part-reasoning module for active-part segmentation and articulation prediction. We further train the model with a geometry-to-articulation curriculum and a decoupled two-pass strategy to balance reconstruction and part-level supervision. On PartNet-Mobility, Artic-O achieves strong reconstruction quality while being substantially more efficient than LARM, a strong prior method. It reduces Chamfer Distance, improves F-score, and achieves comparable or better articulation accuracy across most joint metrics, while reducing inference time from 9 minutes to about 0.3 seconds per object.

RNA function arises from the coupling of nucleotide sequence and three-dimensional structure, motivating their joint design. Coordinating global folding with nucleotide-level detail remains challenging under limited structural supervision. We introduce La-Ribo, a generative framework for RNA sequence-structure co-design via geometry-latent flow matching. La-Ribo retains a sparse phosphate-sugar--base scaffold and encodes nucleotide identity and local conformation in residue-wise latents. A shared flow network generates both jointly, and an RNA-specific decoder then reconstructs all heavy atoms. To expand supervision, we construct a quality-controlled corpus of 168,561 RNA structures, integrating experimental data with predictions from three folding models, including 10,631 MSA-supported structures generated in this work. La-Ribo improves designability and codesignability over the evaluated baselines across sampling budgets and two refolding models, and the same prior supports scaffold-conditioned inverse folding without additional training.

Accurate LiDAR-camera calibration is fundamental to fusing multi-modal perception in autonomous driving and robotic systems. Traditional calibration methods require extensive data collection in controlled environments and cannot compensate for the transformation changes during the vehicle/robot movement. In this paper, we propose the first model that uses bird's-eye view (BEV) features to perform LiDAR camera calibration from raw data, termed BEVCALIB. To achieve this, we extract camera BEV features and LiDAR BEV features separately and fuse them into a shared BEV feature space. To fully utilize the geometric information from the BEV feature, we introduce a novel feature selector to filter the most important features in the transformation decoder, which reduces memory consumption and enables efficient training. Extensive evaluations on KITTI, NuScenes, and our own dataset demonstrate that BEVCALIB establishes a new state of the art. Under various noise conditions, BEVCALIB outperforms the best baseline in the literature by an average of (47.08%, 82.32%) on KITTI dataset, and (78.17%, 68.29%) on NuScenes dataset, in terms of (translation, rotation), respectively. In the open-source domain, it improves the best reproducible baseline by one order of magnitude. Our code and demo results are available at https://cisl.ucr.edu/BEVCalib.

We investigate the relationship between representation geometry and neural network performance. Analyzing 52 pretrained ImageNet models across 13 architecture families, we show that effective dimension -- an unsupervised geometric metric -- strongly predicts accuracy. Output effective dimension achieves partial r=0.75 (p < 10^(-10)) after controlling for model capacity, while total compression achieves partial r=-0.72. These findings replicate across ImageNet and CIFAR-10, and generalize to NLP: effective dimension predicts performance for 8 encoder models on SST-2/MNLI and 15 decoder-only LLMs on AG News (r=0.69, p=0.004), while model size does not (r=0.07). We establish bidirectional causality: degrading geometry via noise causes accuracy loss (r=-0.94, p < 10^(-9)), while improving geometry via PCA maintains accuracy across architectures (-0.03pp at 95% variance). This relationship is noise-type agnostic -- Gaussian, Uniform, Dropout, and Salt-and-pepper noise all show |r| > 0.90. These results establish that effective dimension provides domain-agnostic predictive and causal information about neural network performance, computed entirely without labels.

Vision Language Models (VLMs) have achieved remarkable progress in multimodal tasks, yet they often struggle with visual arithmetic, seemingly simple capabilities like object counting or length comparison, which are essential for relevant complex tasks like chart understanding and geometric reasoning. In this work, we first investigate the root causes of this deficiency through a suite of probing tasks focusing on basic visual arithmetic. Our analysis reveals that while pre-trained vision encoders typically capture sufficient information, the text decoder often fails to decode it correctly for arithmetic reasoning. To address this, we propose CogAlign, a novel post-training strategy inspired by Piaget's theory of cognitive development. CogAlign trains VLMs to recognize invariant properties under visual transformations. We demonstrate that this approach significantly improves the performance of three diverse VLMs on our proposed probing tasks. Furthermore, CogAlign enhances performance by an average of 4.6% on CHOCOLATE and 2.9% on MATH-VISION, outperforming or matching supervised fine-tuning methods while requiring only 60% less training data. These results highlight the effectiveness and generalizability of CogAlign in improving fundamental visual arithmetic capabilities and their transfer to downstream tasks.

The rapid development of Large Multimodal Models (LMMs) has led to remarkable progress in 2D visual understanding; however, extending these capabilities to 3D scene understanding remains a significant challenge. Existing approaches predominantly rely on text-only supervision, which fails to provide the geometric constraints required for learning robust 3D spatial representations. In this paper, we introduce Reg3D, a novel Reconstructive Geometry Instruction Tuning framework that addresses this limitation by incorporating geometry-aware supervision directly into the training process. Our key insight is that effective 3D understanding necessitates reconstructing underlying geometric structures rather than merely describing them. Unlike existing methods that inject 3D information solely at the input level, Reg3D adopts a dual-supervision paradigm that leverages 3D geometric information both as input and as explicit learning targets. Specifically, we design complementary object-level and frame-level reconstruction tasks within a dual-encoder architecture, enforcing geometric consistency to encourage the development of spatial reasoning capabilities. Extensive experiments on ScanQA, Scan2Cap, ScanRefer, and SQA3D demonstrate that Reg3D delivers substantial performance improvements, establishing a new training paradigm for spatially aware multimodal models.

Geometry problem solving has attracted much attention in the NLP community recently. The task is challenging as it requires abstract problem understanding and symbolic reasoning with axiomatic knowledge. However, current datasets are either small in scale or not publicly available. Thus, we construct a new large-scale benchmark, Geometry3K, consisting of 3,002 geometry problems with dense annotation in formal language. We further propose a novel geometry solving approach with formal language and symbolic reasoning, called Interpretable Geometry Problem Solver (Inter-GPS). Inter-GPS first parses the problem text and diagram into formal language automatically via rule-based text parsing and neural object detecting, respectively. Unlike implicit learning in existing methods, Inter-GPS incorporates theorem knowledge as conditional rules and performs symbolic reasoning step by step. Also, a theorem predictor is designed to infer the theorem application sequence fed to the symbolic solver for the more efficient and reasonable searching path. Extensive experiments on the Geometry3K and GEOS datasets demonstrate that Inter-GPS achieves significant improvements over existing methods. The project with code and data is available at https://lupantech.github.io/inter-gps.

Automatic math problem solving has recently attracted increasing attention as a long-standing AI benchmark. In this paper, we focus on solving geometric problems, which requires a comprehensive understanding of textual descriptions, visual diagrams, and theorem knowledge. However, the existing methods were highly dependent on handcraft rules and were merely evaluated on small-scale datasets. Therefore, we propose a Geometric Question Answering dataset GeoQA, containing 4,998 geometric problems with corresponding annotated programs, which illustrate the solving process of the given problems. Compared with another publicly available dataset GeoS, GeoQA is 25 times larger, in which the program annotations can provide a practical testbed for future research on explicit and explainable numerical reasoning. Moreover, we introduce a Neural Geometric Solver (NGS) to address geometric problems by comprehensively parsing multimodal information and generating interpretable programs. We further add multiple self-supervised auxiliary tasks on NGS to enhance cross-modal semantic representation. Extensive experiments on GeoQA validate the effectiveness of our proposed NGS and auxiliary tasks. However, the results are still significantly lower than human performance, which leaves large room for future research. Our benchmark and code are released at https://github.com/chen-judge/GeoQA .

Geometric spatial reasoning forms the foundation of many applications in artificial intelligence, yet the ability of large language models (LLMs) to operate over geometric spatial information expressed in procedural code remains underexplored. In this paper, we address this gap by formalizing the Program-to-Geometry task, which challenges models to translate programmatic drawing code into accurate and abstract geometric reasoning. To evaluate this capability, we present GeoGramBench, a benchmark of 500 carefully refined problems organized by a tailored three-level taxonomy that considers geometric complexity rather than traditional mathematical reasoning complexity. Our comprehensive evaluation of 17 frontier LLMs reveals consistent and pronounced deficiencies: even the most advanced models achieve less than 50% accuracy at the highest abstraction level. These results highlight the unique challenges posed by program-driven spatial reasoning and establish GeoGramBench as a valuable resource for advancing research in symbolic-to-spatial geometric reasoning. Project page: https://github.com/LiAuto-DSR/GeoGramBench.

Large language models learn similar behaviours, yet it remains unclear what structure they share or how to change one behaviour without disturbing others. The Fisher-Rao geometry of next-token probabilities connects these questions: behaviour determines this geometry up to output-preserving symmetries, whereas activation geometry depends on coordinates. Across transformer, state-space and recurrent models, output geometries agree more strongly than activation geometries, and shared geometry supports semantic-category transfer. Agreement with human word choices increases with predictive accuracy, scale and training, and improves further after model-only calibration. Token probabilities and read-out geometry jointly predict the spectrum and its effective dimension. Controlled language assignments show that geometry follows the language law across architectures. Pretraining corpus statistics predict held-out fact acquisition without recalibration, while randomised experiments show that deeper evidence substantially delays acquisition across every tested architecture and evidence construction. Finally, the geometry prescribes minimum-disturbance local interventions, predicts their relative cost, and supports reusable control: updates learned on donor prompts transfer to unseen prompts while better preserving behaviour on reference prompts than Euclidean control. The same geometric correction improves steering, editing, attribution, dictionary learning and fine-tuning.

Multimodal Large Language Models (MLLMs) have achieved remarkable progress but continue to struggle with geometric reasoning, primarily due to the perception bottleneck regarding fine-grained visual elements. While formal languages have aided plane geometry understanding, solid geometry which requires spatial understanding remains largely unexplored. In this paper, we address this challenge by designing a unified formal language that integrates plane and solid geometry, comprehensively covering geometric structures and semantic relations. We construct GDP-29K, a large-scale dataset comprising 20k plane and 9k solid geometry samples collected from diverse real-world sources, each paired with its ground-truth formal description. To ensure syntactic correctness and geometric consistency, we propose a training paradigm that combines Supervised Fine-Tuning with Reinforcement Learning via Verifiable Rewards. Experiments show that our approach achieves state-of-the-art parsing performance. Furthermore, we demonstrate that our parsed formal descriptions serve as a critical cognitive scaffold, significantly boosting MLLMs' capabilities for downstream geometry reasoning tasks. Our data and code are available at Geoparsing.

A reliable language model should be able to signal, prior to generation, when a query falls outside its knowledge. We investigate whether representation geometry can provide such a pre-generation signal by measuring the deviation of hidden states from an answerable reference set, requiring no labeled failure data and no access to model outputs. Across three instruction-tuned models (Llama 3.1-8B, Qwen 2.5-7B, and Mistral-7B-Instruct) and three prompt forms (Math, Fact, Code), we find that geometry primarily encodes task form. Within mathematical prompts, unanswerable inputs consistently deviate from the answerable centroid, yielding strong separation (ROC-AUC 0.78-0.84). This single-pass pre-generation signal outperforms a simple refusal baseline and compares favorably to self-consistency. It also captures cases where models do not explicitly refuse. In contrast, no reliable geometric signal emerges for factual prompts, indicating that the effect is form-conditional rather than universal. Code prompts show large effect sizes with higher variance, suggesting partial generalization beyond mathematical form. A layer-wise analysis reveals that the signal arises in early layers and gradually attenuates toward the output. These results suggest that answerability-related geometry is established before the final stages of generation. Together, these findings indicate that geometric deviation can serve as a lightweight pre-generation signal that is reliable in structured domains with formal answerability constraints, with clear boundaries on where it generalizes.

Recent advances in scene understanding benefit a lot from depth maps because of the 3D geometry information, especially in complex conditions (e.g., low light and overexposed). Existing approaches encode depth maps along with RGB images and perform feature fusion between them to enable more robust predictions. Taking into account that depth can be regarded as a geometry supplement for RGB images, a straightforward question arises: Do we really need to explicitly encode depth information with neural networks as done for RGB images? Based on this insight, in this paper, we investigate a new way to learn RGBD feature representations and present DFormerv2, a strong RGBD encoder that explicitly uses depth maps as geometry priors rather than encoding depth information with neural networks. Our goal is to extract the geometry clues from the depth and spatial distances among all the image patch tokens, which will then be used as geometry priors to allocate attention weights in self-attention. Extensive experiments demonstrate that DFormerv2 exhibits exceptional performance in various RGBD semantic segmentation benchmarks. Code is available at: https://github.com/VCIP-RGBD/DFormer.

We present MoGe, a powerful model for recovering 3D geometry from monocular open-domain images. Given a single image, our model directly predicts a 3D point map of the captured scene with an affine-invariant representation, which is agnostic to true global scale and shift. This new representation precludes ambiguous supervision in training and facilitate effective geometry learning. Furthermore, we propose a set of novel global and local geometry supervisions that empower the model to learn high-quality geometry. These include a robust, optimal, and efficient point cloud alignment solver for accurate global shape learning, and a multi-scale local geometry loss promoting precise local geometry supervision. We train our model on a large, mixed dataset and demonstrate its strong generalizability and high accuracy. In our comprehensive evaluation on diverse unseen datasets, our model significantly outperforms state-of-the-art methods across all tasks, including monocular estimation of 3D point map, depth map, and camera field of view. Code and models can be found on our project page.

Mapping high-fidelity 3D geometry to a representation that allows for intuitive edits remains an elusive goal in computer vision and graphics. The key challenge is the need to model both continuous and discrete shape variations. Current approaches, such as implicit shape representation, lack straightforward interpretable encoding, while others that employ procedural methods output coarse geometry. We present GeoCode, a technique for 3D shape synthesis using an intuitively editable parameter space. We build a novel program that enforces a complex set of rules and enables users to perform intuitive and controlled high-level edits that procedurally propagate at a low level to the entire shape. Our program produces high-quality mesh outputs by construction. We use a neural network to map a given point cloud or sketch to our interpretable parameter space. Once produced by our procedural program, shapes can be easily modified. Empirically, we show that GeoCode can infer and recover 3D shapes more accurately compared to existing techniques and we demonstrate its ability to perform controlled local and global shape manipulations.

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.

Deep neural networks are prone to adversarial examples that maliciously alter the network's outcome. Due to the increasing popularity of 3D sensors in safety-critical systems and the vast deployment of deep learning models for 3D point sets, there is a growing interest in adversarial attacks and defenses for such models. So far, the research has focused on the semantic level, namely, deep point cloud classifiers. However, point clouds are also widely used in a geometric-related form that includes encoding and reconstructing the geometry. In this work, we are the first to consider the problem of adversarial examples at a geometric level. In this setting, the question is how to craft a small change to a clean source point cloud that leads, after passing through an autoencoder model, to the reconstruction of a different target shape. Our attack is in sharp contrast to existing semantic attacks on 3D point clouds. While such works aim to modify the predicted label by a classifier, we alter the entire reconstructed geometry. Additionally, we demonstrate the robustness of our attack in the case of defense, where we show that remnant characteristics of the target shape are still present at the output after applying the defense to the adversarial input. Our code is publicly available at https://github.com/itailang/geometric_adv.

Generating a street-level 3D scene from a single satellite image is a crucial yet challenging task. Current methods present a stark trade-off: geometry-colorization models achieve high geometric fidelity but are typically building-focused and lack semantic diversity. In contrast, proxy-based models use feed-forward image-to-3D frameworks to generate holistic scenes by jointly learning geometry and texture, a process that yields rich content but coarse and unstable geometry. We attribute these geometric failures to the extreme viewpoint gap and sparse, inconsistent supervision inherent in satellite-to-street data. We introduce Sat3DGen to address these fundamental challenges, which embodies a geometry-first methodology. This methodology enhances the feed-forward paradigm by integrating novel geometric constraints with a perspective-view training strategy, explicitly countering the primary sources of geometric error. This geometry-centric strategy yields a dramatic leap in both 3D accuracy and photorealism. For validation, we first constructed a new benchmark by pairing the VIGOR-OOD test set with high-resolution DSM data. On this benchmark, our method improves geometric RMSE from 6.76m to 5.20m. Crucially, this geometric leap also boosts photorealism, reducing the Fréchet Inception Distance (FID) from sim40 to 19 against the leading method, Sat2Density++, despite using no extra tailored image-quality modules. We demonstrate the versatility of our high-quality 3D assets through diverse downstream applications, including semantic-map-to-3D synthesis, multi-camera video generation, large-scale meshing, and unsupervised single-image Digital Surface Model (DSM) estimation. The code has been released on https://github.com/qianmingduowan/Sat3DGen.

Vision-language models (VLM) excel at general understanding yet remain weak at dynamic spatial reasoning (DSR), i.e., reasoning about the evolvement of object geometry and relationship in 3D space over time, largely due to the scarcity of scalable 4D-aware training resources. To bridge this gap across aspects of dataset, benchmark and model, we introduce DSR Suite. First, we propose an automated pipeline that generates multiple-choice question-answer pairs from in-the-wild videos for DSR. By leveraging modern vision foundation models, the pipeline extracts rich geometric and motion information, including camera poses, local point clouds, object masks, orientations, and 3D trajectories. These geometric cues enable the construction of DSR-Train for learning and further human-refined DSR-Bench for evaluation. Compared with previous works, our data emphasize (i) in-the-wild video sources, (ii) object- and scene-level 3D requirements, (iii) viewpoint transformations, (iv) multi-object interactions, and (v) fine-grained, procedural answers. Beyond data, we propose a lightweight Geometry Selection Module (GSM) to seamlessly integrate geometric priors into VLMs, which condenses question semantics and extracts question-relevant knowledge from pretrained 4D reconstruction priors into a compact set of geometry tokens. This targeted extraction avoids overwhelming the model with irrelevant knowledge. Experiments show that integrating DSR-Train and GSM into Qwen2.5-VL-7B significantly enhances its dynamic spatial reasoning capability, while maintaining accuracy on general video understanding benchmarks.

AI-driven geometric problem solving is a complex vision-language task that requires accurate diagram interpretation, mathematical reasoning, and robust cross-modal grounding. A foundational yet underexplored capability for this task is the ability to identify and interpret geometric elements based on natural language queries. To address this, we introduce the task of Referring Expression Comprehension (REC) for geometric problems, which evaluates whether models can localize points, shapes, and spatial relations in diagrams in response to textual prompts. We present GeoRef, a benchmark dataset constructed from existing geometric problem corpora, featuring diverse, high-quality annotations and queries. Due to the lack of annotated data for this task, we generate a large-scale synthetic training dataset using a structured geometric formal language, enabling broad coverage of geometric concepts and facilitating model adaptation. We explore two fine-tuning approaches: Supervised Fine-Tuning (SFT) and Group Relative Policy Optimization (GRPO). Our results show that GRPO significantly outperforms SFT by better aligning model behavior with task-specific rewards. Furthermore, we propose a verify-and-regenerate mechanism that detects incorrect predictions and re-infers answers using contextual reasoning history, further boosting accuracy. Notably, even state-of-the-art Multimodal Large Language Models (MLLMs) struggle with this task, underscoring the necessity of explicitly evaluating and strengthening geometric grounding as a prerequisite for robust geometric problem solving. Moreover, models trained on GeoRef demonstrate measurable improvements on downstream geometric reasoning tasks, highlighting the broader value of REC as a foundation for multimodal mathematical understanding.

This paper presents GPSM4K, a comprehensive geometry multimodal dataset tailored to augment the problem-solving capabilities of Large Vision Language Models (LVLMs). GPSM4K encompasses 2157 multimodal question-answer pairs manually extracted from mathematics textbooks spanning grades 7-12 and is further augmented to 5340 problems, consisting of both numerical and theorem-proving questions. In contrast to PGPS9k, Geometry3K, and Geo170K which feature only objective-type questions, GPSM4K offers detailed step-by-step solutions in a consistent format, facilitating a comprehensive evaluation of problem-solving approaches. This dataset serves as an excellent benchmark for assessing the geometric reasoning capabilities of LVLMs. Evaluation of our test set shows that there is scope for improvement needed in open-source language models in geometry problem-solving. Finetuning on our training set increases the geometry problem-solving capabilities of models. Further, We also evaluate the effectiveness of techniques such as image captioning and Retrieval Augmentation generation (RAG) on model performance. We leveraged LLM to automate the task of final answer evaluation by providing ground truth and predicted solutions. This research will help to assess and improve the geometric reasoning capabilities of LVLMs.

We propose GeoFormer, a Geometry-Aware Transformer architecture specifically designed for prestack seismic data. Unlike Vision Transformer, whose tokens are extracted from 2D patches and primarily encode visual patterns, GeoFormer is designed for prestack seismic data by explicitly incorporating acquisition geometry. Each seismic trace is represented by a 5D unit consisting of the waveform and four source-receiver coordinates, from which two geometric attributes are derived: offset and relative elevation, where relative elevation is the receiver elevation minus the source elevation. GeoFormer therefore performs trace-level tokenization, where each token combines the waveform with these geometric attributes. To exploit these geometric attributes, GeoFormer introduces three geometry injection mechanisms operating at different levels of the Transformer pipeline. At the token level, GeomMLP replaces the classification token with a per-trace geometric representation derived from offset and elevation. At the normalization level, GeomAdaLN replaces uniform layer normalization with geometry-conditioned feature modulation. At the attention level, GeomAttnBias injects a parameter-free physical prior that geometrically proximate traces should attend more to each other. We validate GeoFormer on first-arrival picking, a representative seismic processing task that relies heavily on acquisition geometry. Experiments on four field datasets demonstrate that GeoFormer outperforms Vision Transformer and two task-specific baselines. Crucially, GeoFormer maintains robust picking accuracy under strong noise because its trace-level tokens encode geometric attributes that provide a physical prior independent of waveform quality. Ablation studies confirm that GeomMLP, GeomAdaLN, and GeomAttnBias each contribute to GeoFormer's picking accuracy.

Recovering high-quality depth maps from compressed sources has gained significant attention due to the limitations of consumer-grade depth cameras and the bandwidth restrictions during data transmission. However, current methods still suffer from two challenges. First, bit-depth compression produces a uniform depth representation in regions with subtle variations, hindering the recovery of detailed information. Second, densely distributed random noise reduces the accuracy of estimating the global geometric structure of the scene. To address these challenges, we propose a novel framework, termed geometry-decoupled network (GDNet), for compressed depth map super-resolution that decouples the high-quality depth map reconstruction process by handling global and detailed geometric features separately. To be specific, we propose the fine geometry detail encoder (FGDE), which is designed to aggregate fine geometry details in high-resolution low-level image features while simultaneously enriching them with complementary information from low-resolution context-level image features. In addition, we develop the global geometry encoder (GGE) that aims at suppressing noise and extracting global geometric information effectively via constructing compact feature representation in a low-rank space. We conduct experiments on multiple benchmark datasets, demonstrating that our GDNet significantly outperforms current methods in terms of geometric consistency and detail recovery. In the ECCV 2024 AIM Compressed Depth Upsampling Challenge, our solution won the 1st place award. Our codes are available at: https://github.com/Ian0926/GDNet.

We introduce Thinking with Spatial Code, a framework that transforms RGB video into explicit, temporally coherent 3D representations for physical-world visual question answering. We highlight the empirical finding that our proposed spatial encoder can parse videos into structured spatial code with explicit 3D oriented bounding boxes and semantic labels, enabling large language models (LLMs) to reason directly over explicit spatial variables. Specifically, we propose the spatial encoder that encodes image and geometric features by unifying 6D object parsing and tracking backbones with geometric prediction, and we further finetuning LLMs with reinforcement learning using a spatial rubric reward that encourages perspective-aware, geometrically grounded inference. As a result, our model outperforms proprietary vision-language models on VSI-Bench, setting a new state-of-the-art. Code is available at https://github.com/Beckschen/spatialcode.

Open-world 3D affordance grounding requires localizing functional object parts in 3D given free-form language queries. Existing methods typically assume pre-built object-centric 3D geometry and closed affordance ontologies, limiting deployment from raw RGB observations. We present AffordAny, an end-to-end framework that uses one monocular RGB image to construct large-scale text-conditioned 3D part supervision, ground affordances with a frozen vision-language model (VLM) guided decoder, and improve open-world generalization through pseudo-label self-training. Our automated pipeline produces a benchmark of 5,334 objects and 10,633 part-level samples spanning 473 categories, an order-of-magnitude increase in categorical diversity over prior work. The decoder progressively fuses frozen Cosmos-2B features with 3D geometry through spatial projection, instruction-conditioned semantic compression, and bidirectional geometry-semantics interaction. Minimal-perturbation pseudo-label self-training further adds new objects without human annotation. Under a systematic generalization protocol evaluating unseen objects, unseen categories, and unseen instruction paraphrases, our approach achieves 0.428 IoU on unseen objects and 0.315 IoU on unseen categories after self-training, with unseen-category mIoU improving by 6.3% relative (p<0.01) and an instruction sensitivity gap of only 0.105, demonstrating effectiveness and robustness of our method.

Monocular geometry estimation has recently achieved impressive performance across diverse scenes. However, state-of-the-art models still face notable distortion in local 3D structure, especially in fine details, like thin structures and small objects. We attribute this limitation to an architectural mismatch: most current models decode 3D geometry within a 2D parameterization, where feature interactions are governed by image-plane proximity rather than true 3D spatial relationships. This inadvertently mixes features from geometrically distant surfaces, resulting in over-smoothed geometry particularly around thin or elongated structure. In this paper, we propose MoGe-3, a fine-detail monocular geometry estimation model with Self-Guided Sparse 3D Refinement (SSR) that lifts monocular geometry modeling from 2D image space to 3D space for high-fidelity metric-scale point maps. MoGe-3 lifts the coarse point map from a foundation base model onto a sparse voxel shell and refines it via SSR. The SSR employs sparse convolutions that aggregate features based on 3D spatial locality, avoiding feature mixing across depth discontinuities. Extensive experiments on diverse datasets demonstrate that MoGe-3 significantly outperforms existing approaches in recovering fine detailed 3D geometry across both quantitative metrics and qualitative visualizations. Project page: https://qft-333.github.io/moge3page/

Compact token sequences are essential for efficient 3D generation. However, existing 3D tokenizers typically organize latent representations either over spatial regions or as fixed-size sets of global tokens, both suffering sharp reconstruction degradation when compressed to extremely low token budgets. In this paper, we present ZipTok3D, a 3D tokenizer designed for high-fidelity reconstruction from extremely short token sequences. Its key idea is to organize object geometry into progressively informative global-token prefixes and unfold these compact representations through iterative decoding. Specifically, nested dropout randomly truncates the latent sequence after encoding during training and requires each retained prefix to reconstruct the complete object, thereby prioritizing essential geometric information in the leading tokens. The decoder then repeatedly applies a parameter-shared Transformer block to recover fine-grained geometry from each prefix without a separate generative sampling stage. With the same token dimension, ZipTok3D achieves reconstruction quality comparable to the 32-token COD-VAE baseline using only one token on ShapeNet and four on TRELLIS, yielding 32times and 8times shorter token sequences, respectively.

Large language models have seen widespread adoption in math problem-solving. However, in geometry problems that usually require visual aids for better understanding, even the most advanced multi-modal models currently still face challenges in effectively using image information. High-quality data is crucial for enhancing the geometric capabilities of multi-modal models, yet existing open-source datasets and related efforts are either too challenging for direct model learning or suffer from misalignment between text and images. To overcome this issue, we introduce a novel pipeline that leverages GPT-4 and GPT-4V to generate relatively basic geometry problems with aligned text and images, facilitating model learning. We have produced a dataset of 4.9K geometry problems and combined it with 19K open-source data to form our GeoGPT4V dataset. Experimental results demonstrate that the GeoGPT4V dataset significantly improves the geometry performance of various models on the MathVista and MathVision benchmarks. The code is available at https://github.com/Lanyu0303/GeoGPT4V_Project

Robust 3D representation learning forms the perceptual foundation of spatial intelligence, enabling downstream tasks in scene understanding and embodied AI. However, learning such representations directly from unposed multi-view images remains challenging. Recent self-supervised methods attempt to unify geometry, appearance, and semantics in a feed-forward manner, but they often suffer from weak geometry induction, limited appearance detail, and inconsistencies between geometry and semantics. We introduce UniSplat, a feed-forward framework designed to address these limitations through three complementary components. First, we propose a dual-masking strategy that strengthens geometry induction in the encoder. By masking both encoder and decoder tokens, and targeting decoder masks toward geometry-rich regions, the model is forced to infer structural information from incomplete visual cues, yielding geometry-aware representations even under unposed inputs. Second, we develop a coarse-to-fine Gaussian splatting strategy that reduces appearance-semantics inconsistencies by progressively refining the radiance field. Finally, to enforce geometric-semantic consistency, we introduce a pose-conditioned recalibration mechanism that interrelates the outputs of multiple heads by re-projecting predicted 3D point and semantic maps into the image plane using estimated camera parameters, and aligning them with corresponding RGB and semantic predictions to ensure cross-task consistency, thereby resolving geometry-semantic mismatches. Together, these components yield unified 3D representations that are robust to unposed, sparse-view inputs and generalize across diverse tasks, laying a perceptual foundation for spatial intelligence.

Large Multimodal Models (LMMs) have achieved remarkable success on images and short videos, yet scaling them to long videos remains challenging due to frame-centric tokenization and limited context windows. 3D geometry provides a natural compression mechanism for visual streams: depth and camera pose enable observations from multiple views and time steps to be fused into a persistent, world-aligned representation. While recent 3D LMMs leverage geometry-aware representations to improve spatial reasoning, they continue to lag behind specialist 3D perception systems on grounding and segmentation tasks. We argue that a key limitation is geometry-aware decoding: existing methods communicate 3D predictions through language tokens, proposal selection, or lightweight grounding queries, creating a bottleneck between language reasoning and dense geometric prediction. Building on these insights, we introduce Qwen-3D, a geometry-aware LMM that compresses visual information within the Qwen backbone using multi-view geometric cues, enabling efficient long-horizon visual reasoning over static scenes. Qwen-3D augments visual tokens with 3D Rotary Positional Embeddings, allowing attention to operate directly in 3D scene space rather than across independent image frames and thereby facilitating scalable cross-view and temporal reasoning. To bridge language and geometry, Qwen-3D incorporates a query-based segmentation decoder that grounds language directly in the underlying 3D scene representation, unifying referential grounding, instance segmentation, and visual question answering across both images and videos. Across a diverse set of benchmarks, Qwen-3D surpasses existing 3D LMMs and outperforms several large proprietary 2D models. Notably, Qwen-3D achieves these improvements while maintaining strong performance on standard 2D vision-language benchmarks by jointly training on 2D and 3D data.

Audio language models are designed to understand speech, yet it remains unclear whether they capture how something is said beyond what is said. We present a mechanistic analysis of paralinguistic information in four open source models, Whisper-large-v2, Qwen2-Audio-7B Instruct, Qwen2.5-Omni-7B, and Chroma-4B, using the Expresso dataset with controlled speaking styles. We combine centered kernel alignment, linear probing with leave one speaker out evaluation, open ended tone prediction, and a content prosody leakage metric to trace how style information moves from the audio encoder to the final output. All models strongly encode speaking style in the late encoder, that is, the top third of the audio encoder's layers, but this information is consistently degraded before reaching the output. The projector reshapes representation geometry without removing information, while decoders differ in how much style they preserve depending on architecture and training objective. At the output level, models fall into two behaviors. Some are content driven, where predictions depend mainly on text. Others are acoustic driven, where predictions vary with speaking style. The leakage metric quantifies this difference, and qualitative results confirm it. Overall, we identify a gap between what models encode and what they use, highlighting a key limitation in current audio language models.

Reconstructing and tracking dynamic 3D scenes remains a fundamental challenge in computer vision. Existing approaches often decouple geometry from motion: multi-view reconstruction methods assume static scenes, while dynamic tracking frameworks rely on explicit camera pose estimation or separate motion models. We propose Flow4R, a unified framework that treats camera-space scene flow as the central representation linking 3D structure, object motion, and camera motion. Flow4R predicts a minimal per-pixel property set-3D point position, scene flow, pose weight, and confidence-from two-view inputs using a Vision Transformer. This flow-centric formulation allows local geometry and bidirectional motion to be inferred symmetrically with a shared decoder in a single forward pass, without requiring explicit pose regressors or bundle adjustment. Trained jointly on static and dynamic datasets, Flow4R achieves state-of-the-art performance on 4D reconstruction and tracking tasks, demonstrating the effectiveness of the flow-central representation for spatiotemporal scene understanding.

Chain-of-Thought (CoT) reasoning has become a powerful driver of trajectory prediction in VLA-based autonomous driving, yet its autoregressive nature imposes a latency cost that is prohibitive for real-time deployment. Latent CoT methods attempt to close this gap by compressing reasoning into continuous hidden states, but consistently fall short of their explicit counterparts. We suggest that this is due to purely linguistic latent representations compressing a symbolic abstraction of the world, rather than the causal dynamics that actually govern driving. Thus, we present OneVL (One-step latent reasoning and planning with Vision-Language explanations), a unified VLA and World Model framework that routes reasoning through compact latent tokens supervised by dual auxiliary decoders. Alongside a language decoder that reconstructs text CoT, we introduce a visual world model decoder that predicts future-frame tokens, forcing the latent space to internalize the causal dynamics of road geometry, agent motion, and environmental change. A three-stage training pipeline progressively aligns these latents with trajectory, language, and visual objectives, ensuring stable joint optimization. At inference, the auxiliary decoders are discarded and all latent tokens are prefilled in a single parallel pass, matching the speed of answer-only prediction. Across four benchmarks, OneVL becomes the first latent CoT method to surpass explicit CoT, delivering state-of-the-art accuracy at answer-only latency, and providing direct evidence that tighter compression, when guided in both language and world-model supervision, produces more generalizable representations than verbose token-by-token reasoning. Project Page: https://xiaomi-embodied-intelligence.github.io/OneVL

Boundary representation (B-Rep) generation is a fundamental task in computer-aided design, yet the direct synthesis of high-fidelity and structurally valid B-Reps remains a major challenge. Existing deep generative methods suffer from two forms of brittleness: representation brittleness, caused by padding noise and feature contamination in the latent space, and generation brittleness, stemming from sequential error propagation and a train-inference mismatch due to non-differentiable validity enforcement. We propose HiFi-BRep, a novel framework that addresses these limitations through two synergistic contributions. First, a topology-aware encoder constructs a high-fidelity latent representation by eliminating padding via learnable queries and preventing feature contamination with topology-guided attention. Second, a single-stage decoder jointly predicts geometry and topology in parallel, embedding core manifold constraints as a differentiable learning objective. This design ensures mutual guidance between geometry and topology while avoiding cascaded errors. Extensive experiments show that HiFi-BRep significantly outperforms state-of-the-art methods in both structural validity and geometric fidelity, providing a robust solution for high-quality B-Rep synthesis. Code and models are publicly available at https://github.com/1nnoh/HiFi-BRep.

Single-image feed-forward 3D Gaussian Splatting (3DGS) aims to directly generate a renderable 3D scene representation from one input image, avoiding the cost of multi-view capture and per-scene optimization. However, existing methods are often constrained by a pixel-aligned representation, where Gaussians are predicted from fixed image-grid locations. Such pixel-aligned primitives can produce promising nearby-view renderings, but they remain weakly coupled to underlying scene surfaces and struggle to preserve coherent structures under large viewpoint shifts. We present InfiniSplat, a feed-forward single-image 3DGS framework that moves from a pixel-aligned representation toward a surface-aligned representation. InfiniSplat constructs this representation by first using geometry-guided sampling to place 2D supports according to depth-induced local surface structure, and then applying a query-conditioned implicit decoder to predict Gaussian attributes from the image features queried at these supports.By grounding support locations in geometry while decoupling Gaussian prediction from fixed pixel centers, InfiniSplat produces Gaussian layouts that better follow scene surfaces and reduce scattered primitives caused by grid discretization.Across multiple cross-dataset NVS evaluations, InfiniSplat achieves state-of-the-art performance compared with single-image feed-forward baselines, and demonstrates zero-shot generalization from Hypersim indoor synthetic training to complex open-world scenes.Project page: https://zju3dv.github.io/InfiniSplat.

Recent advances in neural surrogate modeling offer the potential for transformative innovations in applications such as automotive aerodynamics. Yet, industrial-scale problems often involve volumetric meshes with cell counts reaching the 100 millions, presenting major scalability challenges. Complex geometries further complicate modeling through intricate surface-volume interactions, while quantities such as vorticity are highly nonlinear and must satisfy strict divergence-free constraints. To address these requirements, we introduce AB-UPT as a novel modeling scheme for building neural surrogates for CFD simulations. AB-UPT is designed to: (i) decouple geometry encoding and prediction tasks via multi-branch operators; (ii) enable scalability to high-resolution outputs via neural simulation in a low-dimensional latent space, coupled with anchored neural field decoders to predict high-fidelity outputs; (iii) enforce physics consistency by a novel divergence-free formulation. We show that AB-UPT yields state-of-the-art predictive accuracy of surface and volume fields on automotive CFD simulations ranging from 33 thousand up to 150 million mesh cells. Furthermore, our anchored neural field architecture enables the enforcement of hard physical constraints on the physics predictions without degradation in performance, exemplified by modeling divergence-free vorticity fields. Notably, the proposed models can be trained on a single GPU in less than a day and predict industry-standard surface and volume fields within seconds. Additionally, we show that the flexible design of our method enables neural simulation from a CAD geometry alone, omitting the need for costly CFD meshing procedures.

Multi-view camera-based 3D detection is a challenging problem in computer vision. Recent works leverage a pretrained LiDAR detection model to transfer knowledge to a camera-based student network. However, we argue that there is a major domain gap between the LiDAR BEV features and the camera-based BEV features, as they have different characteristics and are derived from different sources. In this paper, we propose Geometry Enhanced Masked Image Modeling (GeoMIM) to transfer the knowledge of the LiDAR model in a pretrain-finetune paradigm for improving the multi-view camera-based 3D detection. GeoMIM is a multi-camera vision transformer with Cross-View Attention (CVA) blocks that uses LiDAR BEV features encoded by the pretrained BEV model as learning targets. During pretraining, GeoMIM's decoder has a semantic branch completing dense perspective-view features and the other geometry branch reconstructing dense perspective-view depth maps. The depth branch is designed to be camera-aware by inputting the camera's parameters for better transfer capability. Extensive results demonstrate that GeoMIM outperforms existing methods on nuScenes benchmark, achieving state-of-the-art performance for camera-based 3D object detection and 3D segmentation. Code and pretrained models are available at https://github.com/Sense-X/GeoMIM.

Feed-forward models for 3D reconstruction have achieved strong performance using deep cross-view attention to exchange information across images. However, these approaches often depend on heavy decoder stacks and lack a structured mechanism for geometry refinement, resulting in poor multi-view consistency. We address this by drawing inspiration from classical bundle adjustment (BA), which can be viewed as an iterative information propagation process between poses and local geometry. Inspired by BA, we propose BA-T, an iterative Transformer that implements BA-style structured updates as a repeatable layer in implicit token space. Instead of relying on deep attention stacks, BA-T refines predictions based on latent residual by a single lightweight layer. Experiments demonstrate that BA-T progressively improves pose and reconstruction accuracy across iterations, achieves stronger cross-view consistency than conventional decoders, and matches or surpasses substantially larger models while using only 16% of their decoder parameters. BA-T provides a compact, efficient, and structural alternative to depth-heavy attention, enabling accurate 3D reconstruction within a lightweight architecture. The code will be made publicly at https://github.com/zhangganlin/BA-T.

Recent works augment Vision-Language Models with geometry features from pretrained 3D models, expecting that the geometric signal will boost spatial reasoning. However, we find that simply fusing geometry features and training on standard spatial QA yields only marginal improvements on high-level multi-hop tasks. We attribute this gap to a training-signal problem: standard spatial QA can be largely answered from visual features and language priors, so the geometry pathway receives weak gradients and fails to integrate with the visual features. To provide a training signal that requires geometry, we propose novel-view semantic rendering as an auxiliary training task that requires the model to predict the semantic layout of an unobserved viewpoint, inspired by humans' ability to mentally simulate novel viewpoints during spatial reasoning. This task encourages joint use of both pathways: geometry provides pose-dependent visibility, while vision provides semantic content. Our auxiliary task yields consistent improvements over the geometry-augmented baseline across all three benchmarks (up to +1.6 on VSI-Bench, +2.2 on ReVSI, +2.9 on our 3D-Point-QA dataset) and our full model surpasses prior open-source methods on VSI-Bench and on ReVSI. Project page: https://yuqunw.github.io/Render2Reason/.

Geometry problem solving (GPS) is a high-level mathematical reasoning requiring the capacities of multi-modal fusion and geometric knowledge application. Recently, neural solvers have shown great potential in GPS but still be short in diagram presentation and modal fusion. In this work, we convert diagrams into basic textual clauses to describe diagram features effectively, and propose a new neural solver called PGPSNet to fuse multi-modal information efficiently. Combining structural and semantic pre-training, data augmentation and self-limited decoding, PGPSNet is endowed with rich knowledge of geometry theorems and geometric representation, and therefore promotes geometric understanding and reasoning. In addition, to facilitate the research of GPS, we build a new large-scale and fine-annotated GPS dataset named PGPS9K, labeled with both fine-grained diagram annotation and interpretable solution program. Experiments on PGPS9K and an existing dataset Geometry3K validate the superiority of our method over the state-of-the-art neural solvers. Our code, dataset and appendix material are available at https://github.com/mingliangzhang2018/PGPS.

Geometry problem-solving (GPS), a challenging task requiring both visual comprehension and symbolic reasoning, effectively measures the reasoning capabilities of multimodal large language models (MLLMs). Humans exhibit strong reasoning ability in this task through accurate identification and adaptive application of geometric principles within visual contexts. However, existing benchmarks fail to jointly assess both dimensions of the human-like geometric reasoning mechanism in MLLMs, remaining a critical gap in assessing their ability to tackle GPS. To this end, we introduce GeoSense, the first comprehensive bilingual benchmark designed to systematically evaluate the geometric reasoning abilities of MLLMs through the lens of geometric principles. GeoSense features a five-level hierarchical framework of geometric principles spanning plane and solid geometry, an intricately annotated dataset of 1,789 problems, and an innovative evaluation strategy. Through extensive experiments on GeoSense with various open-source and closed-source MLLMs, we observe that Gemini-2.0-pro-flash performs best, achieving an overall score of 65.3. Our in-depth analysis reveals that the identification and application of geometric principles remain a bottleneck for leading MLLMs, jointly hindering their reasoning abilities. These findings underscore GeoSense's potential to guide future advancements in MLLMs' geometric reasoning capabilities, paving the way for more robust and human-like reasoning in artificial intelligence.

In this paper, we address the challenging problem of 3D toonification, which involves transferring the style of an artistic domain onto a target 3D face with stylized geometry and texture. Although fine-tuning a pre-trained 3D GAN on the artistic domain can produce reasonable performance, this strategy has limitations in the 3D domain. In particular, fine-tuning can deteriorate the original GAN latent space, which affects subsequent semantic editing, and requires independent optimization and storage for each new style, limiting flexibility and efficient deployment. To overcome these challenges, we propose DeformToon3D, an effective toonification framework tailored for hierarchical 3D GAN. Our approach decomposes 3D toonification into subproblems of geometry and texture stylization to better preserve the original latent space. Specifically, we devise a novel StyleField that predicts conditional 3D deformation to align a real-space NeRF to the style space for geometry stylization. Thanks to the StyleField formulation, which already handles geometry stylization well, texture stylization can be achieved conveniently via adaptive style mixing that injects information of the artistic domain into the decoder of the pre-trained 3D GAN. Due to the unique design, our method enables flexible style degree control and shape-texture-specific style swap. Furthermore, we achieve efficient training without any real-world 2D-3D training pairs but proxy samples synthesized from off-the-shelf 2D toonification models.

3D object segmentation with Large Language Models (LLMs) has become a prevailing paradigm due to its broad semantics, task flexibility, and strong generalization. However, this paradigm is hindered by representation misalignment: LLMs process high-level semantic tokens, whereas 3D point clouds convey only dense geometric structures. In prior methods, misalignment limits both input and output. At the input stage, dense point patches require heavy pre-alignment, weakening object-level semantics and confusing similar distractors. At the output stage, predictions depend only on dense features without explicit geometric cues, leading to a loss of fine-grained accuracy. To address these limitations, we present the Point Linguist Model (PLM), a general framework that bridges the representation gap between LLMs and dense 3D point clouds without requiring large-scale pre-alignment between 3D-text or 3D-images. Specifically, we introduce Object-centric Discriminative Representation (OcDR), which learns object-centric tokens that capture target semantics and scene relations under a hard negative-aware training objective. This mitigates the misalignment between LLM tokens and 3D points, enhances resilience to distractors, and facilitates semantic-level reasoning within LLMs. For accurate segmentation, we introduce the Geometric Reactivation Decoder (GRD), which predicts masks by combining OcDR tokens carrying LLM-inferred geometry with corresponding dense features, preserving comprehensive dense features throughout the pipeline. Extensive experiments show that PLM achieves significant improvements of +7.3 mIoU on ScanNetv2 and +6.0 mIoU on Multi3DRefer for 3D referring segmentation, with consistent gains across 7 benchmarks spanning 4 different tasks, demonstrating the effectiveness of comprehensive object-centric reasoning for robust 3D understanding.

We present TableSeq, an image-only, end-to-end framework for joint table structure recognition, content recognition, and cell localization. The model formulates these tasks as a single sequence-generation problem: one decoder produces an interleaved stream of HTML tags, cell text, and discretized coordinate tokens, thereby aligning logical structure, textual content, and cell geometry within a unified autoregressive sequence. This design avoids external OCR, auxiliary decoders, and complex multi-stage post-processing. TableSeq combines a lightweight high-resolution FCN-H16 encoder with a minimal structure-prior head and a single-layer transformer encoder, yielding a compact architecture that remains effective on challenging layouts. Across standard benchmarks, TableSeq achieves competitive or state-of-the-art results while preserving architectural simplicity. It reaches 95.23 TEDS / 96.83 S-TEDS on PubTabNet, 97.45 TEDS / 98.69 S-TEDS on FinTabNet, and 99.79 / 99.54 / 99.66 precision / recall / F1 on SciTSR under the CAR protocol, while remaining competitive on PubTables-1M under GriTS. Beyond TSR/TCR, the same sequence interface generalizes to index-based table querying without task-specific heads, achieving the best IRDR score and competitive ICDR/ICR performance. We also study multi-token prediction for faster blockwise decoding and show that it reduces inference latency with only limited accuracy degradation. Overall, TableSeq provides a practical and reproducible single-stream baseline for unified table recognition, and the source code will be made publicly available at https://github.com/hamdilaziz/TableSeq.

We present 4RC, a unified feed-forward framework for 4D reconstruction from monocular videos. Unlike existing approaches that typically decouple motion from geometry or produce limited 4D attributes such as sparse trajectories or two-view scene flow, 4RC learns a holistic 4D representation that jointly captures dense scene geometry and motion dynamics. At its core, 4RC introduces a novel encode-once, query-anywhere and anytime paradigm: a transformer backbone encodes the entire video into a compact spatio-temporal latent space, from which a conditional decoder can efficiently query 3D geometry and motion for any query frame at any target timestamp. To facilitate learning, we represent per-view 4D attributes in a minimally factorized form by decomposing them into base geometry and time-dependent relative motion. Extensive experiments demonstrate that 4RC outperforms prior and concurrent methods across a wide range of 4D reconstruction tasks.

Recent vision-language-action (VLA) models and world action models (WAMs) advance robotic manipulation by enriching intermediate representations with auxiliary spatial features or future visual-state prediction. However, these representations largely remain within the observation space and do not share the rigid-body geometry of the action space, forcing the action decoder to implicitly recover this geometry. We propose OASIS, a visuomotor policy that aligns the intermediate representation with the action space via SE(3) end-effector trajectory prediction. OASIS couples a 3D-aware feature encoder that fuses vision-language and metric-depth features with an SE(3) trajectory predictor that produces a camera-frame end-effector trajectory. Conditioned on the predictor's pose-supervised hidden states, the action decoder generates action chunks consistent with rigid-body motion. Across simulation and real-world experiments, OASIS outperforms VLA and WAM baselines in success rate and out-of-distribution generalization. Our project page is available at https://npuhandsome.github.io/OASIS_web.

Geometry diagram parsing plays a key role in geometry problem solving, wherein the primitive extraction and relation parsing remain challenging due to the complex layout and between-primitive relationship. In this paper, we propose a powerful diagram parser based on deep learning and graph reasoning. Specifically, a modified instance segmentation method is proposed to extract geometric primitives, and the graph neural network (GNN) is leveraged to realize relation parsing and primitive classification incorporating geometric features and prior knowledge. All the modules are integrated into an end-to-end model called PGDPNet to perform all the sub-tasks simultaneously. In addition, we build a new large-scale geometry diagram dataset named PGDP5K with primitive level annotations. Experiments on PGDP5K and an existing dataset IMP-Geometry3K show that our model outperforms state-of-the-art methods in four sub-tasks remarkably. Our code, dataset and appendix material are available at https://github.com/mingliangzhang2018/PGDP.

Recently, implicit neural representations have gained popularity for learning-based 3D reconstruction. While demonstrating promising results, most implicit approaches are limited to comparably simple geometry of single objects and do not scale to more complicated or large-scale scenes. The key limiting factor of implicit methods is their simple fully-connected network architecture which does not allow for integrating local information in the observations or incorporating inductive biases such as translational equivariance. In this paper, we propose Convolutional Occupancy Networks, a more flexible implicit representation for detailed reconstruction of objects and 3D scenes. By combining convolutional encoders with implicit occupancy decoders, our model incorporates inductive biases, enabling structured reasoning in 3D space. We investigate the effectiveness of the proposed representation by reconstructing complex geometry from noisy point clouds and low-resolution voxel representations. We empirically find that our method enables the fine-grained implicit 3D reconstruction of single objects, scales to large indoor scenes, and generalizes well from synthetic to real data.

The intrinsic rotation invariance lies at the core of matching point clouds with handcrafted descriptors. However, it is widely despised by recent deep matchers that obtain the rotation invariance extrinsically via data augmentation. As the finite number of augmented rotations can never span the continuous SO(3) space, these methods usually show instability when facing rotations that are rarely seen. To this end, we introduce RoITr, a Rotation-Invariant Transformer to cope with the pose variations in the point cloud matching task. We contribute both on the local and global levels. Starting from the local level, we introduce an attention mechanism embedded with Point Pair Feature (PPF)-based coordinates to describe the pose-invariant geometry, upon which a novel attention-based encoder-decoder architecture is constructed. We further propose a global transformer with rotation-invariant cross-frame spatial awareness learned by the self-attention mechanism, which significantly improves the feature distinctiveness and makes the model robust with respect to the low overlap. Experiments are conducted on both the rigid and non-rigid public benchmarks, where RoITr outperforms all the state-of-the-art models by a considerable margin in the low-overlapping scenarios. Especially when the rotations are enlarged on the challenging 3DLoMatch benchmark, RoITr surpasses the existing methods by at least 13 and 5 percentage points in terms of Inlier Ratio and Registration Recall, respectively.

Image-based relighting of indoor rooms creates an immersive virtual understanding of the space, which is useful for interior design, virtual staging, and real estate. Relighting indoor rooms from a single image is especially challenging due to complex illumination interactions between multiple lights and cluttered objects featuring a large variety in geometrical and material complexity. Recently, generative models have been successfully applied to image-based relighting conditioned on a target image or a latent code, albeit without detailed local lighting control. In this paper, we introduce ScribbleLight, a generative model that supports local fine-grained control of lighting effects through scribbles that describe changes in lighting. Our key technical novelty is an Albedo-conditioned Stable Image Diffusion model that preserves the intrinsic color and texture of the original image after relighting and an encoder-decoder-based ControlNet architecture that enables geometry-preserving lighting effects with normal map and scribble annotations. We demonstrate ScribbleLight's ability to create different lighting effects (e.g., turning lights on/off, adding highlights, cast shadows, or indirect lighting from unseen lights) from sparse scribble annotations.

Spatial intelligence refers to the ability to perceive, reason about, and describe objects and their relationships within three-dimensional environments, forming a foundation for embodied perception and scene understanding. 3D captioning aims to describe 3D scenes in natural language; however, it remains challenging due to the sparsity and irregularity of point clouds and, more critically, the weak grounding and limited out-of-distribution (OOD) generalization of existing captioners across drastically different environments, including indoor and outdoor 3D scenes. To address this challenge, we propose 3D CoCa v2, a generalizable 3D captioning framework that unifies contrastive vision-language learning with 3D caption generation and further improves robustness via test-time search (TTS) without updating the captioner parameters. 3D CoCa v2 builds on a frozen CLIP-based semantic prior, a spatially-aware 3D scene encoder for geometry, and a multimodal decoder jointly optimized with contrastive and captioning objectives, avoiding external detectors or handcrafted proposals. At inference, TTS produces diverse caption candidates and performs reward-guided selection using a compact scene summary. Experiments show improvements over 3D CoCa of +1.50 CIDEr@0.5IoU on ScanRefer and +1.61 CIDEr@0.5IoU on Nr3D, and +3.8 CIDEr@0.25 in zero-shot OOD evaluation on TOD3Cap. Code will be released at https://github.com/AIGeeksGroup/3DCoCav2.

A central goal of visual recognition is to understand objects and scenes from a single image. 2D recognition has witnessed tremendous progress thanks to large-scale learning and general-purpose representations. Comparatively, 3D poses new challenges stemming from occlusions not depicted in the image. Prior works try to overcome these by inferring from multiple views or rely on scarce CAD models and category-specific priors which hinder scaling to novel settings. In this work, we explore single-view 3D reconstruction by learning generalizable representations inspired by advances in self-supervised learning. We introduce a simple framework that operates on 3D points of single objects or whole scenes coupled with category-agnostic large-scale training from diverse RGB-D videos. Our model, Multiview Compressive Coding (MCC), learns to compress the input appearance and geometry to predict the 3D structure by querying a 3D-aware decoder. MCC's generality and efficiency allow it to learn from large-scale and diverse data sources with strong generalization to novel objects imagined by DALLcdotE 2 or captured in-the-wild with an iPhone.

Realistic and editable animal fur reconstruction from multi-view images is challenging due to fine-scale detail, self-occlusion and obfuscation, and, unlike human hair, the lack of animal-fur datasets. Fur usually covers most of an animal's body, with large inter-species and intra-species variability. We present FurE, an efficient strand-based animal fur reconstruction method that recovers a per-strand, editable groom by optimizing a root-conditioned latent field, decoded into strand geometry via a PCA-based decoder. We reconstruct a defurred animal body using local fur-thickness cues from a surface-constrained Gaussian Frosting representation together with part-based priors. We further show that a PCA-based decoder learned from human-hair strand data can alleviate animal-data scarcity while enabling substantially faster optimization. FurE achieves a 10x speedup in strand training over current SOTA dense per-strand optimization while retaining strand fidelity and generalizing across synthetic and real-world sequences, with quantitative and qualitative validation despite the reduction in training time.

Recent progress in spatial reasoning with Multimodal Large Language Models (MLLMs) increasingly leverages geometric priors from 3D encoders. However, most existing integration strategies remain passive: geometry is exposed as a global stream and fused in an indiscriminate manner, which often induces semantic-geometry misalignment and redundant signals. We propose GeoThinker, a framework that shifts the paradigm from passive fusion to active perception. Instead of feature mixing, GeoThinker enables the model to selectively retrieve geometric evidence conditioned on its internal reasoning demands. GeoThinker achieves this through Spatial-Grounded Fusion applied at carefully selected VLM layers, where semantic visual priors selectively query and integrate task-relevant geometry via frame-strict cross-attention, further calibrated by Importance Gating that biases per-frame attention toward task-relevant structures. Comprehensive evaluation results show that GeoThinker sets a new state-of-the-art in spatial intelligence, achieving a peak score of 72.6 on the VSI-Bench. Furthermore, GeoThinker demonstrates robust generalization and significantly improved spatial perception across complex downstream scenarios, including embodied referring and autonomous driving. Our results indicate that the ability to actively integrate spatial structures is essential for next-generation spatial intelligence. Code can be found at https://github.com/Li-Hao-yuan/GeoThinker.

We scale our conventional sub-150M pretraining recipe from 53.5M to 109.7M parameters, holding the method fixed (Qwen3-style decoder with grouped-query attention, RoPE, SwiGLU, RMSNorm, QK-Norm, and a z-loss; FineWeb-Edu data) and changing only the geometry to a deep-and-thin 23-layer x 576-hidden design. The larger model improves across the board -- BLiMP 78.1 -> 81.3, ARC-Easy 51.4 -> 52.5, WikiText-2 byte-perplexity 2.04 -> 1.95 -- and its 81.3% BLiMP essentially matches GPT-X2-125M (81.28) at about 12% fewer parameters. Notably the 110M model achieves this on fewer training tokens (about 8B vs 12B), so the gain is attributable to capacity and depth, not more data. Both models are deliberately conventional; this report is a clean scaling control and the baseline rung (R0) of an ablation study of what further improves models in this regime. An ablation ladder follows: value residuals (R1) and the Muon optimizer (R2) lift ARC-Easy by a cumulative +3.6 (52.5 -> 56.1) at a near-flat BLiMP and are kept; a diverse data blend (R3) and two logit-distillation settings (R4a/R4b) are not kept -- honest negatives. R3 pins ARC-Easy to FineWeb-Edu's educational filtering rather than raw diversity; distillation from a 1.7B teacher can reach the class-leading ARC-Easy (56.99, matching GPT-X2-125M) but only at a perplexity cost that dialing KD down then erases -- so R2 remains the best kept stack.

This paper investigates the fundamental discontinuity between the latest two Segment Anything Models: SAM2 and SAM3. We explain why the expertise in prompt-based segmentation of SAM2 does not transfer to the multimodal concept-driven paradigm of SAM3. SAM2 operates through spatial prompts points, boxes, and masks yielding purely geometric and temporal segmentation. In contrast, SAM3 introduces a unified vision-language architecture capable of open-vocabulary reasoning, semantic grounding, contrastive alignment, and exemplar-based concept understanding. We structure this analysis through five core components: (1) a Conceptual Break Between Prompt-Based and Concept-Based Segmentation, contrasting spatial prompt semantics of SAM2 with multimodal fusion and text-conditioned mask generation of SAM3; (2) Architectural Divergence, detailing pure vision-temporal design of SAM2 versus integration of vision-language encoders, geometry and exemplar encoders, fusion modules, DETR-style decoders, object queries, and ambiguity-handling via Mixture-of-Experts in SAM3; (3) Dataset and Annotation Differences, contrasting SA-V video masks with multimodal concept-annotated corpora of SAM3; (4) Training and Hyperparameter Distinctions, showing why SAM2 optimization knowledge does not apply to SAM3; and (5) Evaluation, Metrics, and Failure Modes, outlining the transition from geometric IoU metrics to semantic, open-vocabulary evaluation. Together, these analyses establish SAM3 as a new class of segmentation foundation model and chart future directions for the emerging concept-driven segmentation era.

Multimodal large language models (MLLMs) have made significant progress in integrating visual and linguistic understanding. Existing benchmarks typically focus on high-level semantic capabilities, such as scene understanding and visual reasoning, but often overlook a crucial, foundational ability: geometric perception. Geometric perception involves understanding geometric shapes, structures, and spatial relationships, which are essential for supporting higher-level semantic tasks. Despite its importance, this capability remains underexplored in current MLLM research. To address this gap, we introduce GePBench, a novel benchmark designed to assess the geometric perception abilities of MLLMs. Our extensive evaluations reveal that current state-of-the-art MLLMs exhibit significant deficiencies in geometric perception tasks. Furthermore, we show that models trained with GePBench data demonstrate substantial improvements on a wide range of benchmark tasks, highlighting the critical role of geometric perception in enabling advanced multimodal applications. Our code and datasets will be publicly available.

We introduce methods for obtaining pretrained Geometric Neural Operators (GNPs) that can serve as basal foundation models for use in obtaining geometric features. These can be used within data processing pipelines for machine learning tasks and numerical methods. We show how our GNPs can be trained to learn robust latent representations for the differential geometry of point-clouds to provide estimates of metric, curvature, and other shape-related features. We demonstrate how our pre-trained GNPs can be used (i) to estimate the geometric properties of surfaces of arbitrary shape and topologies with robustness in the presence of noise, (ii) to approximate solutions of geometric partial differential equations (PDEs) on manifolds, and (iii) to solve equations for shape deformations such as curvature driven flows. We release codes and weights for using GNPs in the package geo_neural_op. This allows for incorporating our pre-trained GNPs as components for reuse within existing and new data processing pipelines. The GNPs also can be used as part of numerical solvers involving geometry or as part of methods for performing inference and other geometric tasks.

Large vision language models exhibit notable limitations on Geometry Problem Solving (GPS) because of their unreliable diagram interpretation and pure natural-language reasoning. A recent line of work mitigates this by using symbolic solvers: the model directly generates a formal program that a geometry solver can execute. However, this direct program generation lacks intermediate reasoning, making the decision process opaque and prone to errors. In this work, we explore a new approach that integrates Chain-of-Thought (CoT) with formal language. The model interleaves natural language reasoning with incremental emission of solver-executable code, producing a hybrid reasoning trace in which critical derivations are expressed in formal language. To teach this behavior at scale, we combine (1) supervised fine-tuning on an 11K newly developed synthetic dataset with interleaved natural language reasoning and automatic formalization, and (2) solver-in-the-loop reinforcement learning that jointly optimizes both the CoT narrative and the resulting program through outcome-based rewards. Built on Qwen2.5-VL-7B, our new model, named GF-Reasoner, achieves up to 15% accuracy improvements on standard GPS benchmarks, surpassing both 7B-scale peers and the much larger model Qwen2.5-VL-72B. By exploiting high-order geometric knowledge and offloading symbolic computation to the solver, the generated reasoning traces are noticeably shorter and cleaner. Furthermore, we present a comprehensive analysis of method design choices (e.g., reasoning paradigms, data synthesis, training epochs, etc.), providing actionable insights for future research.

Existing Vision Language Models (VLMs) architecturally rooted in "flatland" perception, fundamentally struggle to comprehend real-world 3D spatial intelligence. This failure stems from a dual-bottleneck: input-stage conflict between computationally exorbitant geometric-aware encoders and superficial 2D-only features, and output-stage misalignment where discrete tokenizers are structurally incapable of producing precise, continuous numerical values. To break this impasse, we introduce GEODE (Geometric-Output and Decoupled-Input Engine), a novel architecture that resolves this dual-bottleneck by decoupling 3D reasoning from numerical generation. GEODE augments main VLM with two specialized, plug-and-play modules: Decoupled Rationale Module (DRM) that acts as spatial co-processor, aligning explicit 3D data with 2D visual features via cross-attention and distilling spatial Chain-of-Thought (CoT) logic into injectable Rationale Tokens; and Direct Regression Head (DRH), an "Embedding-as-Value" paradigm which routes specialized control tokens to a lightweight MLP for precise, continuous regression of scalars and 3D bounding boxes. The synergy of these modules allows our 1.5B parameter model to function as a high-level semantic dispatcher, achieving state-of-the-art spatial reasoning performance that rivals 7B+ models.

In this paper, we present a novel indoor 3D reconstruction method with occluded surface completion, given a sequence of depth readings. Prior state-of-the-art (SOTA) methods only focus on the reconstruction of the visible areas in a scene, neglecting the invisible areas due to the occlusions, e.g., the contact surface between furniture, occluded wall and floor. Our method tackles the task of completing the occluded scene surfaces, resulting in a complete 3D scene mesh. The core idea of our method is learning 3D geometry prior from various complete scenes to infer the occluded geometry of an unseen scene from solely depth measurements. We design a coarse-fine hierarchical octree representation coupled with a dual-decoder architecture, i.e., Geo-decoder and 3D Inpainter, which jointly reconstructs the complete 3D scene geometry. The Geo-decoder with detailed representation at fine levels is optimized online for each scene to reconstruct visible surfaces. The 3D Inpainter with abstract representation at coarse levels is trained offline using various scenes to complete occluded surfaces. As a result, while the Geo-decoder is specialized for an individual scene, the 3D Inpainter can be generally applied across different scenes. We evaluate the proposed method on the 3D Completed Room Scene (3D-CRS) and iTHOR datasets, significantly outperforming the SOTA methods by a gain of 16.8% and 24.2% in terms of the completeness of 3D reconstruction. 3D-CRS dataset including a complete 3D mesh of each scene is provided at project webpage.

Problems involving geometric data arise in a variety of fields, including computer vision, robotics, chemistry, and physics. Such data can take numerous forms, such as points, direction vectors, planes, or transformations, but to date there is no single architecture that can be applied to such a wide variety of geometric types while respecting their symmetries. In this paper we introduce the Geometric Algebra Transformer (GATr), a general-purpose architecture for geometric data. GATr represents inputs, outputs, and hidden states in the projective geometric algebra, which offers an efficient 16-dimensional vector space representation of common geometric objects as well as operators acting on them. GATr is equivariant with respect to E(3), the symmetry group of 3D Euclidean space. As a transformer, GATr is scalable, expressive, and versatile. In experiments with n-body modeling and robotic planning, GATr shows strong improvements over non-geometric baselines.

Geometry is invariant to viewpoint, which makes any collection of images a redundant encoding of a single 3D state. Existing feed-forward reconstruction models fail to exploit this: per-view methods emit overlapping, unaligned pointmaps that grow linearly with input count, while global-latent methods commit to a fixed, low-resolution output. We introduce Surflo, which compresses a variable number of unposed RGB views into K latent tokens-one global state-and decodes oriented 3D surface points by independently transporting them from noise onto the surface via flow matching. This frees the output from any fixed grid or token budget: the same latent yields from a few thousand to a million points in a single forward pass. To suppress the local inconsistencies inherent to independent per-point decoding, an inference-time guidance term correlates nearby points by injecting a photometric gradient during ODE integration. Surflo matches or surpasses feed-forward baselines on surface metrics, runs an order of magnitude faster than optimization-based methods that require hundreds of views, and is the only feed-forward approach to combine a global latent with arbitrary-resolution decoding.

Deep sequence models are said to store atomic facts predominantly in the form of associative memory: a brute-force lookup of co-occurring entities. We identify a dramatically different form of storage of atomic facts that we term as geometric memory. Here, the model has synthesized embeddings encoding novel global relationships between all entities, including ones that do not co-occur in training. Such storage is powerful: for instance, we show how it transforms a hard reasoning task involving an ell-fold composition into an easy-to-learn 1-step navigation task. From this phenomenon, we extract fundamental aspects of neural embedding geometries that are hard to explain. We argue that the rise of such a geometry, as against a lookup of local associations, cannot be straightforwardly attributed to typical supervisory, architectural, or optimizational pressures. Counterintuitively, a geometry is learned even when it is more complex than the brute-force lookup. Then, by analyzing a connection to Node2Vec, we demonstrate how the geometry stems from a spectral bias that -- in contrast to prevailing theories -- indeed arises naturally despite the lack of various pressures. This analysis also points out to practitioners a visible headroom to make Transformer memory more strongly geometric. We hope the geometric view of parametric memory encourages revisiting the default intuitions that guide researchers in areas like knowledge acquisition, capacity, discovery, and unlearning.

Multimodal large language models (MLLMs) have made rapid progress in recent years, yet continue to struggle with low-level visual perception (LLVP) -- particularly the ability to accurately describe the geometric details of an image. This capability is crucial for applications in areas such as robotics, medical image analysis, and manufacturing. In this paper, we first introduce Geoperception, a benchmark designed to evaluate an MLLM's ability to accurately transcribe 2D geometric information from an image. Using this benchmark, we demonstrate the limitations of leading MLLMs, and then conduct a comprehensive empirical study to explore strategies for improving their performance on geometric tasks. Our findings highlight the benefits of certain model architectures, training techniques, and data strategies, including the use of high-fidelity synthetic data and multi-stage training with a data curriculum. Notably, we find that a data curriculum enables models to learn challenging geometry understanding tasks which they fail to learn from scratch. Leveraging these insights, we develop Euclid, a family of models specifically optimized for strong low-level geometric perception. Although purely trained on synthetic multimodal data, Euclid shows strong generalization ability to novel geometry shapes. For instance, Euclid outperforms the best closed-source model, Gemini-1.5-Pro, by up to 58.56% on certain Geoperception benchmark tasks and 10.65% on average across all tasks.

Empowered by large-scale training, vision-language models (VLMs) achieve strong image and video understanding, yet their ability to perform spatial reasoning in both static scenes and dynamic videos remains limited. Recent advances try to handle this limitation by injecting geometry tokens from pretrained 3D foundation models into VLMs. Nevertheless, we observe that naive token fusion followed by standard fine-tuning in this line of work often leaves such geometric cues underutilized for spatial reasoning, as VLMs tend to rely heavily on 2D visual cues. In this paper, we propose GeoSR, a framework designed to make geometry matter by encouraging VLMs to actively reason with geometry tokens. GeoSR introduces two key components: (1) Geometry-Unleashing Masking, which strategically masks portions of 2D vision tokens during training to weaken non-geometric shortcuts and force the model to consult geometry tokens for spatial reasoning; and (2) Geometry-Guided Fusion, a gated routing mechanism that adaptively amplifies geometry token contributions in regions where geometric evidence is critical. Together, these designs unleash the potential of geometry tokens for spatial reasoning tasks. Extensive experiments on both static and dynamic spatial reasoning benchmarks demonstrate that GeoSR consistently outperforms prior methods and establishes new state-of-the-art performance by effectively leveraging geometric information. The project page is available at https://suhzhang.github.io/GeoSR/.

Existing Large Multimodal Models (LMMs) struggle with mathematical geometric reasoning due to a lack of high-quality image-text paired data. Current geometric data generation approaches, which apply preset templates to generate geometric data or use Large Language Models (LLMs) to rephrase questions and answers (Q&A), unavoidably limit data accuracy and diversity. To synthesize higher-quality data, we propose a two-stage Reverse Chain-of-Thought (R-CoT) geometry problem generation pipeline. First, we introduce GeoChain to produce high-fidelity geometric images and corresponding descriptions highlighting relations among geometric elements. We then design a Reverse A&Q method that reasons step-by-step based on the descriptions and generates questions in reverse from the reasoning results. Experiments demonstrate that the proposed method brings significant and consistent improvements on multiple LMM baselines, achieving new performance records in the 2B, 7B, and 8B settings. Notably, R-CoT-8B significantly outperforms previous state-of-the-art open-source mathematical models by 16.6% on MathVista and 9.2% on GeoQA, while also surpassing the closed-source model GPT-4o by an average of 13% across both datasets. The code is available at https://github.com/dle666/R-CoT.

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.

Perceiving and reconstructing 4D spatial-temporal geometry from videos is a fundamental yet challenging computer vision task. To facilitate interactive and real-time applications, we propose a streaming 4D visual geometry transformer that shares a similar philosophy with autoregressive large language models. We explore a simple and efficient design and employ a causal transformer architecture to process the input sequence in an online manner. We use temporal causal attention and cache the historical keys and values as implicit memory to enable efficient streaming long-term 4D reconstruction. This design can handle real-time 4D reconstruction by incrementally integrating historical information while maintaining high-quality spatial consistency. For efficient training, we propose to distill knowledge from the dense bidirectional visual geometry grounded transformer (VGGT) to our causal model. For inference, our model supports the migration of optimized efficient attention operator (e.g., FlashAttention) from the field of large language models. Extensive experiments on various 4D geometry perception benchmarks demonstrate that our model increases the inference speed in online scenarios while maintaining competitive performance, paving the way for scalable and interactive 4D vision systems. Code is available at: https://github.com/wzzheng/StreamVGGT.

Towards intelligent image editing, object removal should eliminate both the target object and its causal visual artifacts, such as shadows and reflections. However, existing image appearance-based methods either follow strictly mask-aligned training and fail to remove these causal effects which are not explicitly masked, or adopt loosely mask-aligned strategies that lack controllability and may unintentionally over-erase other objects. We identify that these limitations stem from ignoring the causal relationship between an object's geometry presence and its visual effects. To address this limitation, we propose a geometry-aware two-stage framework that decouples object removal into (1) geometry removal and (2) appearance rendering. In the first stage, we remove the object directly from the geometry (e.g., depth) using strictly mask-aligned supervision, enabling structure-aware editing with strong geometric constraints. In the second stage, we render a photorealistic RGB image conditioned on the updated geometry, where causal visual effects are considered implicitly as a result of the modified 3D geometry. To guide learning in the geometry removal stage, we introduce a preference-driven objective based on positive and negative sample pairs, encouraging the model to remove objects as well as their causal visual artifacts while avoiding new structural insertions. Extensive experiments demonstrate that our method achieves state-of-the-art performance in removing both objects and their associated artifacts on two popular benchmarks. The code is available at https://github.com/buxiangzhiren/GeoRemover.

Feed-forward 3D reconstruction models are efficient but rigid: once trained, they perform inference in a zero-shot manner and cannot adapt to the test scene. As a result, visually plausible reconstructions often contain errors, particularly under occlusions, specularities, and ambiguous cues. To address this, we introduce Free Geometry, a framework that enables feed-forward 3D reconstruction models to self-evolve at test time without any 3D ground truth. Our key insight is that, when the model receives more views, it produces more reliable and view-consistent reconstructions. Leveraging this property, given a testing sequence, we mask a subset of frames to construct a self-supervised task. Free Geometry enforces cross-view feature consistency between representations from full and partial observations, while maintaining the pairwise relations implied by the held-out frames. This self-supervision allows for fast recalibration via lightweight LoRA updates, taking less than 2 minutes per dataset on a single GPU. Our approach consistently improves state-of-the-art foundation models, including Depth Anything 3 and VGGT, across 4 benchmark datasets, yielding an average improvement of 3.73% in camera pose accuracy and 2.88% in point map prediction. Code is available at https://github.com/hiteacherIamhumble/Free-Geometry .

Reconstructing 3D objects into editable programs is pivotal for applications like reverse engineering and shape editing. However, existing methods often rely on limited domain-specific languages (DSLs) and small-scale datasets, restricting their ability to model complex geometries and structures. To address these challenges, we introduce MeshCoder, a novel framework that reconstructs complex 3D objects from point clouds into editable Blender Python scripts. We develop a comprehensive set of expressive Blender Python APIs capable of synthesizing intricate geometries. Leveraging these APIs, we construct a large-scale paired object-code dataset, where the code for each object is decomposed into distinct semantic parts. Subsequently, we train a multimodal large language model (LLM) that translates 3D point cloud into executable Blender Python scripts. Our approach not only achieves superior performance in shape-to-code reconstruction tasks but also facilitates intuitive geometric and topological editing through convenient code modifications. Furthermore, our code-based representation enhances the reasoning capabilities of LLMs in 3D shape understanding tasks. Together, these contributions establish MeshCoder as a powerful and flexible solution for programmatic 3D shape reconstruction and understanding.

Recent advances in large language models (LLMs) have demonstrated remarkable capabilities across diverse domains, particularly in mathematical reasoning, amid which geometry problem solving remains a challenging area where auxiliary construction plays a enssential role. Existing approaches either achieve suboptimal performance or rely on massive LLMs (e.g., GPT-4o), incurring massive computational costs. We posit that reinforcement learning with verifiable reward (e.g., GRPO) offers a promising direction for training smaller models that effectively combine auxiliary construction with robust geometric reasoning. However, directly applying GRPO to geometric reasoning presents fundamental limitations due to its dependence on unconditional rewards, which leads to indiscriminate and counterproductive auxiliary constructions. To address these challenges, we propose Group Contrastive Policy Optimization (GCPO), a novel reinforcement learning framework featuring two key innovations: (1) Group Contrastive Masking, which adaptively provides positive or negative reward signals for auxiliary construction based on contextual utility, and a (2) length reward that promotes longer reasoning chains. Building on GCPO, we develop GeometryZero, a family of affordable-size geometric reasoning models that judiciously determine when to employ auxiliary construction. Our extensive empirical evaluation across popular geometric benchmarks (Geometry3K, MathVista) demonstrates that GeometryZero models consistently outperform baselines (e.g. GRPO), achieving an average improvement of 4.29% across all benchmarks.

Existing RGB-based imitation learning approaches typically employ traditional vision encoders such as ResNet or ViT, which lack explicit 3D reasoning capabilities. Recent geometry-grounded vision models, such as VGGT~wang2025vggt, provide robust spatial understanding and are promising candidates to address this limitation. This work investigates the integration of geometry-aware visual representations into robotic manipulation. Our results suggest that incorporating the geometry-aware vision encoder into imitation learning frameworks, including ACT and DP, yields up to 6.5% improvement over standard vision encoders in success rate across single- and bi-manual manipulation tasks in both simulation and real-world settings. Despite these benefits, most geometry-grounded models require high computational cost, limiting their deployment in practical robotic systems. To address this challenge, we propose eVGGT, an efficient geometry-aware encoder distilled from VGGT. eVGGT is nearly 9 times faster and 5 times smaller than VGGT, while preserving strong 3D reasoning capabilities. Code and pretrained models will be released to facilitate further research in geometry-aware robotics.

Large language models (LLMs) have shown remarkable proficiency in human-level reasoning and generation capabilities, which encourages extensive research on their application in mathematical problem solving. However, current work has been largely focused on text-based mathematical problems, with limited investigation in problems involving geometric information. Addressing this gap, we aim to enable LLMs to solve geometric problems by understanding image input. We first analyze the limitations of current Multimodal Large Language Models (MLLMs) in this area: they struggle to accurately comprehending basic geometric elements and their relationships. To overcome these challenges, we take advantage of the unique characteristics of geometric problems (such as unique geometric logical form, and geometric scalability) and the capacity of the textual LLMs to build an enriched multimodal geometry dataset based on existing data. The augmented dataset, Geo170K, contains more than 170K geometric image-caption and question-answer pairs. Utilizing our constructed Geo170K dataset, we develop G-LLaVA, which demonstrates exceptional performance in solving geometric problems, significantly outperforming GPT-4-V on the MathVista benchmark with only 7B parameters.

Depth estimation and Gaussian reconstruction share a need for multi-scale feature processing, yet use different prediction interfaces and training objectives. Whether a decoder's internal computation can be optimized once and reused across these systems remains a structural design question. Search over primitive configurations alone leaves the organization of transformations fixed. To address this gap, we propose GeoBridge, an evolutionary framework for transferable decoder topologies. First, GeoBridge makes stage-internal topology searchable through residual, sequential, and parallel composition of configurable primitives. Shape-preserving composition retains the stage interfaces of a fixed multi-scale scaffold. Second, a functional-tree encoding supports stage-aware crossover and hierarchical mutation, while a gradient sensitivity-alignment proxy evaluates candidates without training. Multi-objective evolution balances this proxy with computation and parameter cost. Third, selected topologies are instantiated and trained in target systems with task-specific prediction interfaces. A single source search takes approximately 12 GPU hours. GeoBridgeNet-A3 achieves the best depth accuracy among four adapted decoder baselines under a common encoder and training protocol on NYU Depth V2 and KITTI. The same topology reaches 27.11 dB PSNR in Gaussian novel-view synthesis on RealEstate10K, and its refined representations support open-vocabulary segmentation on LERF. These results support decoder topology as a reusable design object across the evaluated geometric vision systems. Code and pretrained weights are available at https://github.com/EMI-Group/GeoBridge-NAS.

Graph foundation models represent a transformative paradigm for learning transferable representations across diverse graph domains. Recent methods leverage large language models to unify graph and text modalities into a shared representation space using contrastive learning. However, systematic evaluations reveal significant performance degradation at structural boundaries where distinct topological patterns converge, with accuracy losses exceeding 20 percentage points. This issue arises from a key limitation: current methods assume all graph structures can be encoded within a single Euclidean space. In reality, tree structures require hyperbolic geometry to preserve hierarchical branching, while cyclic patterns depend on spherical geometry for closure properties. At structural boundaries, nodes experience conflicting geometric constraints that uniform encoding spaces cannot resolve. This raises a crucial challenge: Can alignment frameworks be designed to respect the intrinsic geometric diversity of graph structures? We introduce GraphShaper, a geometry-aware framework that enhances graph encoding through multi-geometric specialization. Our approach employs expert networks tailored to different geometric spaces, dynamically computing fusion weights to adaptively integrate geometric properties based on local structural characteristics. This adaptive fusion preserves structural integrity before alignment with text embeddings. Extensive experiments demonstrate that GraphShaper achieves 9.47\% accuracy improvements on citation networks and 7.63\% on social networks in zero-shot settings.

This is the first paper in a series of work we have accomplished over the past three years. In this paper, we have constructed a consistent formal plane geometry system. This will serve as a crucial bridge between IMO-level plane geometry challenges and readable AI automated reasoning. Within this formal framework, we have been able to seamlessly integrate modern AI models with our formal system. AI is now capable of providing deductive reasoning solutions to IMO-level plane geometry problems, just like handling other natural languages, and these proofs are readable, traceable, and verifiable. We propose the geometry formalization theory (GFT) to guide the development of the geometry formal system. Based on the GFT, we have established the FormalGeo, which consists of 88 geometric predicates and 196 theorems. It can represent, validate, and solve IMO-level geometry problems. we also have crafted the FGPS (formal geometry problem solver) in Python. It serves as both an interactive assistant for verifying problem-solving processes and an automated problem solver. We've annotated the formalgeo7k and formalgeo-imo datasets. The former contains 6,981 (expand to 133,818 through data augmentation) geometry problems, while the latter includes 18 (expand to 2,627 and continuously increasing) IMO-level challenging geometry problems. All annotated problems include detailed formal language descriptions and solutions. Implementation of the formal system and experiments validate the correctness and utility of the GFT. The backward depth-first search method only yields a 2.42% problem-solving failure rate, and we can incorporate deep learning techniques to achieve lower one. The source code of FGPS and datasets are available at https://github.com/BitSecret/FGPS.

Despite remarkable advancements in video depth estimation, existing methods exhibit inherent limitations in achieving geometric fidelity through the affine-invariant predictions, limiting their applicability in reconstruction and other metrically grounded downstream tasks. We propose GeometryCrafter, a novel framework that recovers high-fidelity point map sequences with temporal coherence from open-world videos, enabling accurate 3D/4D reconstruction, camera parameter estimation, and other depth-based applications. At the core of our approach lies a point map Variational Autoencoder (VAE) that learns a latent space agnostic to video latent distributions for effective point map encoding and decoding. Leveraging the VAE, we train a video diffusion model to model the distribution of point map sequences conditioned on the input videos. Extensive evaluations on diverse datasets demonstrate that GeometryCrafter achieves state-of-the-art 3D accuracy, temporal consistency, and generalization capability.

Compressed short-text generators can fail in two different places: the codec may discard information before generation starts, or the latent generator may produce weak codes. Without separating these failure modes, researchers can spend compute improving the wrong component. We study this problem in a controlled 64-to-16 TinyStories case study built from a hierarchical VQ-VAE-2 codec and a masked discrete diffusion generator (MDLM). We use a staged validation protocol that separates codec reconstruction fidelity, latent generation quality, and auxiliary latent diagnostics under one shared external GPT-2 scorer, while reporting complementary semantic metrics for the geometry study. In the tested configuration, codec reconstruction alone raises median external perplexity from 15.17 to 27.36 (+80.4%) and p95 from 25.10 to 98.91 (+294.1%), showing that the dominant quality loss appears before latent generation begins. Under the same scorer, code-space MDLM remains materially stronger than token-space diffusion, reducing mean, median, and p95 by 32.9%, 30.9%, and 36.6%, respectively. Geometry-aware regularization improves local latent proxies but does not improve decoded-text metrics in the available runs. The contribution is methodological rather than algorithmic: the paper presents a reusable staged diagnosis for one concrete pipeline and shows that, in this setting, codec fidelity rather than latent denoising sets the practical quality ceiling.

We present X-Decoder, a generalized decoding model that can predict pixel-level segmentation and language tokens seamlessly. X-Decodert takes as input two types of queries: (i) generic non-semantic queries and (ii) semantic queries induced from text inputs, to decode different pixel-level and token-level outputs in the same semantic space. With such a novel design, X-Decoder is the first work that provides a unified way to support all types of image segmentation and a variety of vision-language (VL) tasks. Further, our design enables seamless interactions across tasks at different granularities and brings mutual benefits by learning a common and rich pixel-level visual-semantic understanding space, without any pseudo-labeling. After pretraining on a mixed set of a limited amount of segmentation data and millions of image-text pairs, X-Decoder exhibits strong transferability to a wide range of downstream tasks in both zero-shot and finetuning settings. Notably, it achieves (1) state-of-the-art results on open-vocabulary segmentation and referring segmentation on eight datasets; (2) better or competitive finetuned performance to other generalist and specialist models on segmentation and VL tasks; and (3) flexibility for efficient finetuning and novel task composition (e.g., referring captioning and image editing). Code, demo, video, and visualization are available at https://x-decoder-vl.github.io.

We present NoReGeo, a novel benchmark designed to evaluate the intrinsic geometric understanding of large language models (LLMs) without relying on reasoning or algebraic computation. Unlike existing benchmarks that primarily assess models' proficiency in reasoning-based geometry-where solutions are derived using algebraic methods-NoReGeo focuses on evaluating whether LLMs can inherently encode spatial relationships and recognize geometric properties directly. Our benchmark comprises 2,500 trivial geometric problems spanning 25 categories, each carefully crafted to be solvable purely through native geometric understanding, assuming known object locations. We assess a range of state-of-the-art models on NoReGeo, including frontier models like GPT-4, observing that even the most advanced systems achieve an overall maximum of 65% accuracy in binary classification tasks. Further, our ablation experiments demonstrate that such geometric understanding does not emerge through fine-tuning alone, indicating that effective training for geometric comprehension requires a specialized approach from the outset. Our findings highlight a significant gap in current LLMs' ability to natively grasp geometric concepts, providing a foundation for future research toward models with true geometric cognition.

We present GenesisGeo, an automated theorem prover in Euclidean geometry. We have open-sourced a large-scale geometry dataset of 21.8 million geometric problems, over 3 million of which contain auxiliary constructions. Specially, we significantly accelerate the symbolic deduction engine DDARN by 120x through theorem matching, combined with a C++ implementation of its core components. Furthermore, we build our neuro-symbolic prover, GenesisGeo, upon Qwen3-0.6B-Base, which solves 24 of 30 problems (IMO silver medal level) in the IMO-AG-30 benchmark using a single model, and achieves 26 problems (IMO gold medal level) with a dual-model ensemble.

In this work, we introduce CAD-Coder, a novel framework that reformulates text-to-CAD as the generation of CadQuery scripts - a Python-based, parametric CAD language. This representation enables direct geometric validation, a richer modeling vocabulary, and seamless integration with existing LLMs. To further enhance code validity and geometric fidelity, we propose a two-stage learning pipeline: (1) supervised fine-tuning on paired text-CadQuery data, and (2) reinforcement learning with Group Reward Policy Optimization (GRPO), guided by a CAD-specific reward comprising both a geometric reward (Chamfer Distance) and a format reward. We also introduce a chain-of-thought (CoT) planning process to improve model reasoning, and construct a large-scale, high-quality dataset of 110K text-CadQuery-3D model triplets and 1.5K CoT samples via an automated pipeline. Extensive experiments demonstrate that CAD-Coder enables LLMs to generate diverse, valid, and complex CAD models directly from natural language, advancing the state of the art of text-to-CAD generation and geometric reasoning.

Geometric diagrams are critical in conveying mathematical and scientific concepts, yet traditional diagram generation methods are often manual and resource-intensive. While text-to-image generation has made strides in photorealistic imagery, creating accurate geometric diagrams remains a challenge due to the need for precise spatial relationships and the scarcity of geometry-specific datasets. This paper presents MagicGeo, a training-free framework for generating geometric diagrams from textual descriptions. MagicGeo formulates the diagram generation process as a coordinate optimization problem, ensuring geometric correctness through a formal language solver, and then employs coordinate-aware generation. The framework leverages the strong language translation capability of large language models, while formal mathematical solving ensures geometric correctness. We further introduce MagicGeoBench, a benchmark dataset of 220 geometric diagram descriptions, and demonstrate that MagicGeo outperforms current methods in both qualitative and quantitative evaluations. This work provides a scalable, accurate solution for automated diagram generation, with significant implications for educational and academic applications.

Transformers achieve strong performance across diverse domains but implicitly assume Euclidean geometry in their attention mechanisms, limiting their effectiveness on data with non-Euclidean structure. While recent extensions to hyperbolic and spherical spaces show promise for hierarchical and cyclical patterns, respectively, they require committing to a single geometry a priori, reducing flexibility when data exhibits mixed geometric properties. We introduce the Curvature-Adaptive Transformer (CAT), a novel architecture that dynamically learns per-token routing across three geometric attention branches through a lightweight, differentiable gating mechanism. Unlike fixed-geometry approaches, CAT enables adaptive geometric specialization, routing tokens to the appropriate curvature based on their local relational structure. The routing network provides interpretable curvature preferences while each branch employs geometry-specific operations optimized for its respective manifold. On knowledge graph completion benchmarks (FB15k-237, WN18RR), CAT achieves approximately 10% improvements in MRR and Hits@10 over fixed-geometry baselines with minimal overhead (5% parameter increase, comparable inference time). These results demonstrate that learned geometric adaptation outperforms any single fixed geometry for complex relational reasoning, establishing CAT as a scalable and interpretable foundation for mixture-of-geometry architectures across language, vision, and multimodal domains.

As transformers are equivariant to the permutation of input tokens, encoding the positional information of tokens is necessary for many tasks. However, since existing positional encoding schemes have been initially designed for NLP tasks, their suitability for vision tasks, which typically exhibit different structural properties in their data, is questionable. We argue that existing positional encoding schemes are suboptimal for 3D vision tasks, as they do not respect their underlying 3D geometric structure. Based on this hypothesis, we propose a geometry-aware attention mechanism that encodes the geometric structure of tokens as relative transformation determined by the geometric relationship between queries and key-value pairs. By evaluating on multiple novel view synthesis (NVS) datasets in the sparse wide-baseline multi-view setting, we show that our attention, called Geometric Transform Attention (GTA), improves learning efficiency and performance of state-of-the-art transformer-based NVS models without any additional learned parameters and only minor computational overhead.

Geometric Problem Solving (GPS) remains at the heart of enhancing mathematical reasoning in large language models because it requires the combination of diagrammatic understanding, symbolic manipulation and logical inference. In existing literature, researchers have chiefly focused on synchronising the diagram descriptions with text literals and solving the problem. In this vein, they have either taken a neural, symbolic or neuro-symbolic approach. But this solves only the first two of the requirements, namely diagrammatic understanding and symbolic manipulation, while leaving logical inference underdeveloped. The logical inference is often limited to one chain-of-thought (CoT). To address this weakness in hitherto existing models, this paper proposes MARS-GPS, that generates multiple parallel reasoning rollouts augmented with Python code execution for numerical verification, ranks them using token-level entropy as a confidence signal, and aggregates answers through a multi-stage voting and self-verification pipeline. Empirical results show that MARS-GPS with 8 parallel rollouts achieves 88.8% on Geometry3K, a nearly +11% improvement over the prior state-of-the-art, with accuracy scaling consistently as the number of rollouts increases from 1 to 16 (+6.0% on ablation subset). We provide our code and data in an anonymous repository: https://anonymous.4open.science/r/MARS-GPS-DE55.