Daily Papers

Real-world spatial intelligence requires agents to understand scenes from continuous video streams, where objects move, persist, disappear, and reappear over time. While recent spatial foundation models have enabled generalizable feed-forward 3D reconstruction, most streaming methods remain geometry-centric and lack temporally consistent object-level understanding. Meanwhile, existing semantic reconstruction and 3D-aware vision-language methods largely rely on externally extracted 2D semantic cues or loosely coupled geometry inputs, limiting unified geometry-instance learning in long dynamic scenes. In this paper, we propose IGGT4D, a streaming instance-grounded geometry Transformer for online 4D scene understanding. IGGT4D processes video frames sequentially, reuses historical context through causal spatial-temporal modeling, and incrementally updates a unified representation of camera motion, geometry, and object identity. This enables long-sequence feed-forward reconstruction with geometry-instance consistency in dynamic environments. To address the lack of high-quality 4D supervision, we further construct InsScene4D-147K, a large-scale dataset spanning real/synthetic and static/dynamic scenes, with RGB images, depth, poses, and temporally consistent instance masks generated by an automated geometry-guided annotation pipeline. Experiments on 3D reconstruction, pose estimation, instance spatial tracking, and open-vocabulary segmentation demonstrate that IGGT4D outperforms existing streaming baselines while maintaining scalable online inference for long dynamic sequences.

Achieving meticulous segmentation of tooth point clouds from intra-oral scans stands as an indispensable prerequisite for various orthodontic applications. Given the labor-intensive nature of dental annotation, a significant amount of data remains unlabeled, driving increasing interest in semi-supervised approaches. One primary challenge of existing semi-supervised medical segmentation methods lies in noisy pseudo labels generated for unlabeled data. To address this challenge, we propose GeoT, the first framework that employs instance-dependent transition matrix (IDTM) to explicitly model noise in pseudo labels for semi-supervised dental segmentation. Specifically, to handle the extensive solution space of IDTM arising from tens of thousands of dental points, we introduce tooth geometric priors through two key components: point-level geometric regularization (PLGR) to enhance consistency between point adjacency relationships in 3D and IDTM spaces, and class-level geometric smoothing (CLGS) to leverage the fixed spatial distribution of tooth categories for optimal IDTM estimation. Extensive experiments performed on the public Teeth3DS dataset and private dataset demonstrate that our method can make full utilization of unlabeled data to facilitate segmentation, achieving performance comparable to fully supervised methods with only 20% of the labeled data.

Accurate 3D instance segmentation is crucial for high-quality scene understanding in the 3D vision domain. However, 3D instance segmentation based on 2D-to-3D lifting approaches struggle to produce precise instance-level segmentation, due to accumulated errors introduced during the lifting process from ambiguous semantic guidance and insufficient depth constraints. To tackle these challenges, we propose splitting and growing reliable semantic mask for high-fidelity 3D instance segmentation (SGS-3D), a novel "split-then-grow" framework that first purifies and splits ambiguous lifted masks using geometric primitives, and then grows them into complete instances within the scene. Unlike existing approaches that directly rely on raw lifted masks and sacrifice segmentation accuracy, SGS-3D serves as a training-free refinement method that jointly fuses semantic and geometric information, enabling effective cooperation between the two levels of representation. Specifically, for semantic guidance, we introduce a mask filtering strategy that leverages the co-occurrence of 3D geometry primitives to identify and remove ambiguous masks, thereby ensuring more reliable semantic consistency with the 3D object instances. For the geometric refinement, we construct fine-grained object instances by exploiting both spatial continuity and high-level features, particularly in the case of semantic ambiguity between distinct objects. Experimental results on ScanNet200, ScanNet++, and KITTI-360 demonstrate that SGS-3D substantially improves segmentation accuracy and robustness against inaccurate masks from pre-trained models, yielding high-fidelity object instances while maintaining strong generalization across diverse indoor and outdoor environments. Code is available in the supplementary materials.

Vision-Language-Action models have achieved remarkable progress in robotic manipulation, yet they suffer from a critical limitation: a lack of 3D scene understanding. This deficiency manifests as three intertwined challenges: weak extraction of 3D spatial positions without enforcing multi-view consistency, inadequate 3D instance understanding, and fragile reasoning under occlusion. Although mature 3D perception methods exist, their direct integration into VLA pipelines is hindered by architectural incompatibility and by heavy reliance on costly instance-level annotations. To address the above challenges, we propose 3DVLA, a plug-and-play framework that injects robust 3D reasoning into pretrained VLAs without requiring extra manual labels or discarding VLM priors. Specifically, 3DVLA tackles the three challenges through: (1) pervasive 3D feature encoding with explicit multi-view consistency constraints across all modalities and a Spatially-Conditioned Geometry Aggregation method, (2) an instance estimation module with high-level instance tokens for 3D instance awareness, and (3) a masked self-supervised 3D encoding branch that retains its predictor for visual token completion to handle occlusions. We integrate 3DVLA with multiple VLA baselines and evaluate on LIBERO-Plus and RoboTwin 2.0. Results show consistent and significant gains in manipulation performance, validating both the effectiveness and plug-and-play compatibility of our approach.

Vision-language models solve geometry problems with rising accuracy, yet their intermediate states remain latent and unverifiable: a relation expressed in textual reasoning or drawing code carries no guarantee that a constraint-satisfying configuration realizes it. We observe that existing externalization methods based on rendered pixels or one-shot scripts fail to provide exact, per-action geometric guarantees. Enforcing geometric relations by algebraic definition closes this gap: the workspace becomes a constraint-checked evolving canvas. We present Draw2Think, a framework that recasts geometric reasoning from latent spatial inference into agentic interaction with the GeoGebra constraint engine. In a Propose-Draw-Verify loop, Draw2Think externalizes hypotheses onto an executable canvas, measures exact geometric quantities, and feeds structured observations back to the model, so subsequent reasoning proceeds from checked canvas state grounded by the shared workspace. This externalization makes two properties separately auditable: model-level Construction Fidelity (whether the canvas realizes the intended configuration) and engine-level Measurement Faithfulness (exact values and relations from canvas constraints). Across construction, outcome, and rendering evaluations, Draw2Think builds canvases that pass 95.9% predicate-level and 84.0% strict problem-level construction checks on GeoGoal, improves outcome accuracy by up to 4.1%/16.4% on planar/solid benchmarks, and attains 68.2%/90.5% strict/relaxed rendering scores on GenExam-math. Project page is available at https://draw2think.github.io/

Large-scale video diffusion models achieve impressive visual quality, yet often fail to preserve geometric consistency. Prior approaches improve consistency either by augmenting the generator with additional modules or applying geometry-aware alignment. However, architectural modifications can compromise the generalization of internet-scale pretrained models, while existing alignment methods are limited to static scenes and rely on RGB-space rewards that require repeated VAE decoding, incurring substantial compute overhead and failing to generalize to highly dynamic real-world scenes. To preserve the pretrained capacity while improving geometric consistency, we propose VGGRPO (Visual Geometry GRPO), a latent geometry-guided framework for geometry-aware video post-training. VGGRPO introduces a Latent Geometry Model (LGM) that stitches video diffusion latents to geometry foundation models, enabling direct decoding of scene geometry from the latent space. By constructing LGM from a geometry model with 4D reconstruction capability, VGGRPO naturally extends to dynamic scenes, overcoming the static-scene limitations of prior methods. Building on this, we perform latent-space Group Relative Policy Optimization with two complementary rewards: a camera motion smoothness reward that penalizes jittery trajectories, and a geometry reprojection consistency reward that enforces cross-view geometric coherence. Experiments on both static and dynamic benchmarks show that VGGRPO improves camera stability, geometry consistency, and overall quality while eliminating costly VAE decoding, making latent-space geometry-guided reinforcement an efficient and flexible approach to world-consistent video generation.

Video diffusion models lack explicit geometric supervision during training, leading to inconsistency artifacts such as object deformation, spatial drift, and depth violations in generated videos. To address this limitation, we propose a geometry-based reward model that leverages pretrained geometric foundation models to evaluate multi-view consistency through cross-frame reprojection error. Unlike previous geometric metrics that measure inconsistency in pixel space, where pixel intensity may introduce additional noise, our approach conducts error computation in a pointwise fashion, yielding a more physically grounded and robust error metric. Furthermore, we introduce a geometry-aware sampling strategy that filters out low-texture and non-semantic regions, focusing evaluation on geometrically meaningful areas with reliable correspondences to improve robustness. We apply this reward model to align video diffusion models through two complementary pathways: post-training of a bidirectional model via SFT or Reinforcement Learning and inference-time optimization of a Causal Video Model (e.g., Streaming video generator) via test-time scaling with our reward as a path verifier. Experimental results validate the effectiveness of our design, demonstrating that our geometry-based reward provides superior robustness compared to other variants. By enabling efficient inference-time scaling, our method offers a practical solution for enhancing open-source video models without requiring extensive computational resources for retraining.

Humans naturally perceive the geometric structure and semantic content of a 3D world as intertwined dimensions, enabling coherent and accurate understanding of complex scenes. However, most prior approaches prioritize training large geometry models for low-level 3D reconstruction and treat high-level spatial understanding in isolation, overlooking the crucial interplay between these two fundamental aspects of 3D-scene analysis, thereby limiting generalization and leading to poor performance in downstream 3D understanding tasks. Recent attempts have mitigated this issue by simply aligning 3D models with specific language models, thus restricting perception to the aligned model's capacity and limiting adaptability to downstream tasks. In this paper, we propose InstanceGrounded Geometry Transformer (IGGT), an end-to-end large unified transformer to unify the knowledge for both spatial reconstruction and instance-level contextual understanding. Specifically, we design a 3D-Consistent Contrastive Learning strategy that guides IGGT to encode a unified representation with geometric structures and instance-grounded clustering through only 2D visual inputs. This representation supports consistent lifting of 2D visual inputs into a coherent 3D scene with explicitly distinct object instances. To facilitate this task, we further construct InsScene-15K, a large-scale dataset with high-quality RGB images, poses, depth maps, and 3D-consistent instance-level mask annotations with a novel data curation pipeline.

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.

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

We tackle the problem of generating a complete vector map representation from aerial imagery in a single run: producing polygons for all land-cover classes with shared boundaries and without gaps or overlaps. Existing polygonization methods are typically class-specific; extending them to multiple classes via per-class runs commonly leads to topological inconsistencies, such as duplicated edges, gaps, and overlaps. We formalize this new task as All-Class Polygonal Vectorization (ACPV) and release the first public benchmark, Deventer-512, with standardized metrics jointly evaluating semantic fidelity, geometric accuracy, vertex efficiency, per-class topological fidelity and global topological consistency. To realize ACPV, we propose ACPV-Net, a unified framework introducing a novel Semantically Supervised Conditioning (SSC) mechanism coupling semantic perception with geometric primitive generation, along with a topological reconstruction that enforces shared-edge consistency by design. While enforcing such strict topological constraints, ACPV-Net surpasses all class-specific baselines in polygon quality across classes on Deventer-512. It also applies to single-class polygonal vectorization without any architectural modification, achieving the best-reported results on WHU-Building. Data, code, and models will be released at: https://github.com/HeinzJiao/ACPV-Net.

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.

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

We introduce the Multi-Instance Generation (MIG) task, which focuses on generating multiple instances within a single image, each accurately placed at predefined positions with attributes such as category, color, and shape, strictly following user specifications. MIG faces three main challenges: avoiding attribute leakage between instances, supporting diverse instance descriptions, and maintaining consistency in iterative generation. To address attribute leakage, we propose the Multi-Instance Generation Controller (MIGC). MIGC generates multiple instances through a divide-and-conquer strategy, breaking down multi-instance shading into single-instance tasks with singular attributes, later integrated. To provide more types of instance descriptions, we developed MIGC++. MIGC++ allows attribute control through text \& images and position control through boxes \& masks. Lastly, we introduced the Consistent-MIG algorithm to enhance the iterative MIG ability of MIGC and MIGC++. This algorithm ensures consistency in unmodified regions during the addition, deletion, or modification of instances, and preserves the identity of instances when their attributes are changed. We introduce the COCO-MIG and Multimodal-MIG benchmarks to evaluate these methods. Extensive experiments on these benchmarks, along with the COCO-Position benchmark and DrawBench, demonstrate that our methods substantially outperform existing techniques, maintaining precise control over aspects including position, attribute, and quantity. Project page: https://github.com/limuloo/MIGC.

We address the problem of detecting and erasing furniture from a wide angle photograph of a room. Inpainting large regions of an indoor scene often results in geometric inconsistencies of background elements within the inpaint mask. To address this problem, we utilize perceptual information (e.g. instance segmentation, and room layout) to produce a geometrically consistent empty version of a room. We share important details to make this system viable, such as per-plane inpainting, automatic rectification, and texture refinement. We provide detailed ablation along with qualitative examples, justifying our design choices. We show an application of our system by removing real furniture from a room and redecorating it with virtual furniture.

Reinforcement learning from human feedback (RLHF) for 3D generation is now established across a number of works, but most existing pipelines optimise explicit surface representations, often by converting radiance fields into meshes and training heavily on surface-supervised data. We instead fine-tune a pretrained 3D-aware generative model directly from a learned reward over radiance-field density (σ) values, with no externally supplied mesh or shape prior. The reward model requires no pretraining, trains easily on a small set of preference samples, and yields robust improvement in 3D geometry. Working on an unconditional 3D-aware face GAN (EG3D), our reward reads the continuous 3D density field of the neural radiance field (NeRF) directly and supplies a geometry-only learning signal, requiring neither text conditioning, mesh extraction, nor multi-view rendering. A density-consistency constraint keeps the 2D appearance qualitatively similar while the geometry is reshaped, at a measurable but bounded distributional cost (FID-50k rises from 4.09 to 6.66): the fine-tuned generator, trained from the preferences of a single annotator as a proof of concept, produces face geometries preferred by users in 74.4% of pairwise comparisons.

Open-vocabulary 3D object detection aims to localize and recognize objects beyond a fixed training taxonomy. In multi-view RGB settings, recent approaches often decouple geometry-based instance construction from semantic labeling, generating class-agnostic fragments and assigning open-vocabulary categories post hoc. While flexible, such decoupling leaves instance construction governed primarily by geometric consistency, without semantic constraints during merging. When geometric evidence is view-dependent and incomplete, this geometry-only merging can lead to irreversible association errors, including over-merging of distinct objects or fragmentation of a single instance. We propose Group3D, a multi-view open-vocabulary 3D detection framework that integrates semantic constraints directly into the instance construction process. Group3D maintains a scene-adaptive vocabulary derived from a multimodal large language model (MLLM) and organizes it into semantic compatibility groups that encode plausible cross-view category equivalence. These groups act as merge-time constraints: 3D fragments are associated only when they satisfy both semantic compatibility and geometric consistency. This semantically gated merging mitigates geometry-driven over-merging while absorbing multi-view category variability. Group3D supports both pose-known and pose-free settings, relying only on RGB observations. Experiments on ScanNet and ARKitScenes demonstrate that Group3D achieves state-of-the-art performance in multi-view open-vocabulary 3D detection, while exhibiting strong generalization in zero-shot scenarios. The project page is available at https://ubin108.github.io/Group3D/.

Representational similarity analysis and related methods compare the internal geometries of neural networks, but they measure only alignment between spaces, leaving a blind spot -- whether a representation's structure is reliably recoverable, not merely similar. We introduce geometric stability, a distinct axis, and Shesha, a metric that quantifies it from a single representation by correlating dissimilarity matrices built from complementary random halves of the feature dimensions. Unlike CKA and Procrustes distance, Shesha is provably non-invariant to orthogonal rotations of the feature basis. This is by design: the basis is privileged for learned models, since probes, patching, and steering act on coordinates, and a rotation-invariant metric cannot see whether the targeted structure survives them. A double dissociation isolates the mechanism -- removing the top principal component collapses CKA while Shesha holds, whereas rotating a representation into its eigenbasis, which preserves the spectrum and CKA exactly, collapses Shesha. Across 2,463 encoder configurations in seven domains, the metrics are redundant under geometry-preserving transforms and anti-correlate under compression (ρ=-0.47). Across 170 vision models spanning 6 clean and 38 corruption-shifted datasets, DINOv2 ranks first or second in transferability on three of six clean datasets yet bottom-quartile in stability on five, an isolated dissociation rather than a trade-off.

The scarcity of annotations poses a significant challenge in medical image analysis. Large-scale pre-training has emerged as a promising label-efficient solution, owing to the utilization of large-scale data, large models, and advanced pre-training techniques. However, its development in medical images remains underexplored. The primary challenge lies in harnessing large-scale unlabeled data and learning high-level semantics without annotations. We observe that 3D medical images exhibit consistent geometric context, i.e., consistent geometric relations between different organs, which leads to a promising way for learning consistent representations. Motivated by this, we introduce a simple-yet-effective Volume Contrast (VoCo) framework to leverage geometric context priors for self-supervision. Given an input volume, we extract base crops from different regions to construct positive and negative pairs for contrastive learning. Then we predict the contextual position of a random crop by contrasting its similarity to the base crops. In this way, VoCo encodes the inherent geometric context into model representations, facilitating high-level semantic learning without annotations. Specifically, we (1) introduce the largest medical pre-training dataset PreCT-160K; (2) investigate scaling laws and propose guidelines for tailoring different model sizes to various medical tasks; (3) build a benchmark encompassing 48 medical tasks. Extensive experiments highlight the superiority of VoCo. Codes at https://github.com/Luffy03/Large-Scale-Medical.

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 .

Automated theorem proving in Euclidean geometry, particularly for International Mathematical Olympiad (IMO) level problems, remains a major challenge and an important research focus in Artificial Intelligence. In this paper, we present a highly efficient method for geometry theorem proving that runs entirely on CPUs without relying on neural network-based inference. Our initial study shows that a simple random strategy for adding auxiliary points can achieve silver-medal level human performance on IMO. Building on this, we propose HAGeo, a Heuristic-based method for adding Auxiliary constructions in Geometric deduction that solves 28 of 30 problems on the IMO-30 benchmark, achieving gold-medal level performance and surpassing AlphaGeometry, a competitive neural network-based approach, by a notable margin. To evaluate our method and existing approaches more comprehensively, we further construct HAGeo-409, a benchmark consisting of 409 geometry problems with human-assessed difficulty levels. Compared with the widely used IMO-30, our benchmark poses greater challenges and provides a more precise evaluation, setting a higher bar for geometry theorem proving.

Large-scale text-to-image models enable a wide range of image editing techniques, using text prompts or even spatial controls. However, applying these editing methods to multi-view images depicting a single scene leads to 3D-inconsistent results. In this work, we focus on spatial control-based geometric manipulations and introduce a method to consolidate the editing process across various views. We build on two insights: (1) maintaining consistent features throughout the generative process helps attain consistency in multi-view editing, and (2) the queries in self-attention layers significantly influence the image structure. Hence, we propose to improve the geometric consistency of the edited images by enforcing the consistency of the queries. To do so, we introduce QNeRF, a neural radiance field trained on the internal query features of the edited images. Once trained, QNeRF can render 3D-consistent queries, which are then softly injected back into the self-attention layers during generation, greatly improving multi-view consistency. We refine the process through a progressive, iterative method that better consolidates queries across the diffusion timesteps. We compare our method to a range of existing techniques and demonstrate that it can achieve better multi-view consistency and higher fidelity to the input scene. These advantages allow us to train NeRFs with fewer visual artifacts, that are better aligned with the target geometry.

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.

The topic of stitching images with globally natural structures holds paramount significance. Current methodologies exhibit the ability to preserve local geometric structures, yet fall short in maintaining relationships between these geometric structures. In this paper, we endeavor to safeguard the overall, OBJect-level structures within images based on Global Similarity Prior, while concurrently mitigating distortion and ghosting artifacts with OBJ-GSP. Our approach leverages the Segment Anything Model to extract geometric structures with semantic information, enhancing the algorithm's ability to preserve objects in a manner that aligns more intuitively with human perception. We seek to identify spatial constraints that govern the relationships between various geometric boundaries. Recognizing that multiple geometric boundaries collectively define complete objects, we employ triangular meshes to safeguard not only individual geometric structures but also the overall shapes of objects within the images. Empirical evaluations across multiple image stitching datasets demonstrate that our method establishes a new state-of-the-art benchmark in image stitching. Our implementation and dataset is publicly available at https://github.com/RussRobin/OBJ-GSP .

Model merging aims to combine multiple models into one without additional training. Naïve parameter-space averaging can be fragile under architectural symmetries, as their geometry does not take them into account. In this work we show that not only the geometry, but also the averaging procedure itself, must be symmetry-invariant to achieve symmetry-aware merges. Consequently, we propose a general solution: merging as Fréchet averaging, i.e., selecting parameters that minimize a sum of geodesic distances on an appropriate manifold. In this view, the key design choice is the overall geometry, i.e., the choice of metric, manifold, and distance approximation, that determines what it means for two models to be "close". We show that Fréchet averaging, combined with simplifying assumptions, contains Fisher merging. Building on this, we examine the particular case of low-rank adapters (LoRA), whose symmetries induce a distinct geometry: that of a quotient manifold. We outline the limitations of current LoRA merging methods, propose a practical algorithm for this setting, and show how they compare with other commonly used approaches.

Large diffusion models demonstrate remarkable zero-shot capabilities in novel view synthesis from a single image. However, these models often face challenges in maintaining consistency across novel and reference views. A crucial factor leading to this issue is the limited utilization of contextual information from reference views. Specifically, when there is an overlap in the viewing frustum between two views, it is essential to ensure that the corresponding regions maintain consistency in both geometry and appearance. This observation leads to a simple yet effective approach, where we propose to use epipolar geometry to locate and retrieve overlapping information from the input view. This information is then incorporated into the generation of target views, eliminating the need for training or fine-tuning, as the process requires no learnable parameters. Furthermore, to enhance the overall consistency of generated views, we extend the utilization of epipolar attention to a multi-view setting, allowing retrieval of overlapping information from the input view and other target views. Qualitative and quantitative experimental results demonstrate the effectiveness of our method in significantly improving the consistency of synthesized views without the need for any fine-tuning. Moreover, This enhancement also boosts the performance of downstream applications such as 3D reconstruction. The code is available at https://github.com/botaoye/ConsisSyn.

The recently developed Sora model [1] has exhibited remarkable capabilities in video generation, sparking intense discussions regarding its ability to simulate real-world phenomena. Despite its growing popularity, there is a lack of established metrics to evaluate its fidelity to real-world physics quantitatively. In this paper, we introduce a new benchmark that assesses the quality of the generated videos based on their adherence to real-world physics principles. We employ a method that transforms the generated videos into 3D models, leveraging the premise that the accuracy of 3D reconstruction is heavily contingent on the video quality. From the perspective of 3D reconstruction, we use the fidelity of the geometric constraints satisfied by the constructed 3D models as a proxy to gauge the extent to which the generated videos conform to real-world physics rules. Project page: https://sora-geometrical-consistency.github.io/

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.

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.

Geometry problems are a crucial testbed for AI reasoning capabilities. Most existing geometry solving systems cannot express problems within a unified framework, thus are difficult to integrate with other mathematical fields. Besides, since most geometric proofs rely on intuitive diagrams, verifying geometry problems is particularly challenging. To address these gaps, we introduce LeanGeo, a unified formal system for formalizing and solving competition-level geometry problems within the Lean 4 theorem prover. LeanGeo features a comprehensive library of high-level geometric theorems with Lean's foundational logic, enabling rigorous proof verification and seamless integration with Mathlib. We also present LeanGeo-Bench, a formal geometry benchmark in LeanGeo, comprising problems from the International Mathematical Olympiad (IMO) and other advanced sources. Our evaluation demonstrates the capabilities and limitations of state-of-the-art Large Language Models on this benchmark, highlighting the need for further advancements in automated geometric reasoning. We open source the theorem library and the benchmark of LeanGeo at https://github.com/project-numina/LeanGeo/tree/master.

Reconstructing 3D objects from a single image guided by pretrained diffusion models has demonstrated promising outcomes. However, due to utilizing the case-agnostic rigid strategy, their generalization ability to arbitrary cases and the 3D consistency of reconstruction are still poor. In this work, we propose Consistent123, a case-aware two-stage method for highly consistent 3D asset reconstruction from one image with both 2D and 3D diffusion priors. In the first stage, Consistent123 utilizes only 3D structural priors for sufficient geometry exploitation, with a CLIP-based case-aware adaptive detection mechanism embedded within this process. In the second stage, 2D texture priors are introduced and progressively take on a dominant guiding role, delicately sculpting the details of the 3D model. Consistent123 aligns more closely with the evolving trends in guidance requirements, adaptively providing adequate 3D geometric initialization and suitable 2D texture refinement for different objects. Consistent123 can obtain highly 3D-consistent reconstruction and exhibits strong generalization ability across various objects. Qualitative and quantitative experiments show that our method significantly outperforms state-of-the-art image-to-3D methods. See https://Consistent123.github.io for a more comprehensive exploration of our generated 3D assets.

Geometric alignment appears in a variety of applications, ranging from domain adaptation, optimal transport, and normalizing flows in machine learning; optical flow and learned augmentation in computer vision and deformable registration within biomedical imaging. A recurring challenge is the alignment of domains whose topology is not the same; a problem that is routinely ignored, potentially introducing bias in downstream analysis. As a first step towards solving such alignment problems, we propose an unsupervised algorithm for the detection of changes in image topology. The model is based on a conditional variational auto-encoder and detects topological changes between two images during the registration step. We account for both topological changes in the image under spatial variation and unexpected transformations. Our approach is validated on two tasks and datasets: detection of topological changes in microscopy images of cells, and unsupervised anomaly detection brain imaging.

Recently, methods leveraging diffusion model priors to assist monocular geometric estimation (e.g., depth and normal) have gained significant attention due to their strong generalization ability. However, most existing works focus on estimating geometric properties within the camera coordinate system of individual video frames, neglecting the inherent ability of diffusion models to determine inter-frame correspondence. In this work, we demonstrate that, through appropriate design and fine-tuning, the intrinsic consistency of video generation models can be effectively harnessed for consistent geometric estimation. Specifically, we 1) select geometric attributes in the global coordinate system that share the same correspondence with video frames as the prediction targets, 2) introduce a novel and efficient conditioning method by reusing positional encodings, and 3) enhance performance through joint training on multiple geometric attributes that share the same correspondence. Our results achieve superior performance in predicting global geometric attributes in videos and can be directly applied to reconstruction tasks. Even when trained solely on static video data, our approach exhibits the potential to generalize to dynamic video scenes.

Learning about the three-dimensional world from two-dimensional images is a fundamental problem in computer vision. An ideal neural network architecture for such tasks would leverage the fact that objects can be rotated and translated in three dimensions to make predictions about novel images. However, imposing SO(3)-equivariance on two-dimensional inputs is difficult because the group of three-dimensional rotations does not have a natural action on the two-dimensional plane. Specifically, it is possible that an element of SO(3) will rotate an image out of plane. We show that an algorithm that learns a three-dimensional representation of the world from two dimensional images must satisfy certain geometric consistency properties which we formulate as SO(2)-equivariance constraints. We use the induced and restricted representations of SO(2) on SO(3) to construct and classify architectures which satisfy these geometric consistency constraints. We prove that any architecture which respects said consistency constraints can be realized as an instance of our construction. We show that three previously proposed neural architectures for 3D pose prediction are special cases of our construction. We propose a new algorithm that is a learnable generalization of previously considered methods. We test our architecture on three pose predictions task and achieve SOTA results on both the PASCAL3D+ and SYMSOL pose estimation tasks.

Matching cost aggregation plays a fundamental role in learning-based multi-view stereo networks. However, directly aggregating adjacent costs can lead to suboptimal results due to local geometric inconsistency. Related methods either seek selective aggregation or improve aggregated depth in the 2D space, both are unable to handle geometric inconsistency in the cost volume effectively. In this paper, we propose GoMVS to aggregate geometrically consistent costs, yielding better utilization of adjacent geometries. More specifically, we correspond and propagate adjacent costs to the reference pixel by leveraging the local geometric smoothness in conjunction with surface normals. We achieve this by the geometric consistent propagation (GCP) module. It computes the correspondence from the adjacent depth hypothesis space to the reference depth space using surface normals, then uses the correspondence to propagate adjacent costs to the reference geometry, followed by a convolution for aggregation. Our method achieves new state-of-the-art performance on DTU, Tanks & Temple, and ETH3D datasets. Notably, our method ranks 1st on the Tanks & Temple Advanced benchmark.

It is inherently ambiguous to lift 2D results from pre-trained diffusion models to a 3D world for text-to-3D generation. 2D diffusion models solely learn view-agnostic priors and thus lack 3D knowledge during the lifting, leading to the multi-view inconsistency problem. We find that this problem primarily stems from geometric inconsistency, and avoiding misplaced geometric structures substantially mitigates the problem in the final outputs. Therefore, we improve the consistency by aligning the 2D geometric priors in diffusion models with well-defined 3D shapes during the lifting, addressing the vast majority of the problem. This is achieved by fine-tuning the 2D diffusion model to be viewpoint-aware and to produce view-specific coordinate maps of canonically oriented 3D objects. In our process, only coarse 3D information is used for aligning. This "coarse" alignment not only resolves the multi-view inconsistency in geometries but also retains the ability in 2D diffusion models to generate detailed and diversified high-quality objects unseen in the 3D datasets. Furthermore, our aligned geometric priors (AGP) are generic and can be seamlessly integrated into various state-of-the-art pipelines, obtaining high generalizability in terms of unseen shapes and visual appearance while greatly alleviating the multi-view inconsistency problem. Our method represents a new state-of-the-art performance with an 85+% consistency rate by human evaluation, while many previous methods are around 30%. Our project page is https://sweetdreamer3d.github.io/

Recent advances in discriminative and generative pretraining have yielded geometry estimation models with strong generalization capabilities. While discriminative monocular geometry estimation methods rely on large-scale fine-tuning data to achieve zero-shot generalization, several generative-based paradigms show the potential of achieving impressive generalization performance on unseen scenes by leveraging pre-trained diffusion models and fine-tuning on even a small scale of synthetic training data. Frustratingly, these models are trained with different recipes on different datasets, making it hard to find out the critical factors that determine the evaluation performance. Besides, current geometry evaluation benchmarks have two main drawbacks that may prevent the development of the field, i.e., limited scene diversity and unfavorable label quality. To resolve the above issues, (1) we build fair and strong baselines in a unified codebase for evaluating and analyzing the geometry estimation models; (2) we evaluate monocular geometry estimators on more challenging benchmarks for geometry estimation task with diverse scenes and high-quality annotations. Our results reveal that pre-trained using large data, discriminative models such as DINOv2, can outperform generative counterparts with a small amount of high-quality synthetic data under the same training configuration, which suggests that fine-tuning data quality is a more important factor than the data scale and model architecture. Our observation also raises a question: if simply fine-tuning a general vision model such as DINOv2 using a small amount of synthetic depth data produces SOTA results, do we really need complex generative models for depth estimation? We believe this work can propel advancements in geometry estimation tasks as well as a wide range of downstream applications.

Given a set of dissimilarity measurements amongst data points, determining what metric representation is most "consistent" with the input measurements or the metric that best captures the relevant geometric features of the data is a key step in many machine learning algorithms. Existing methods are restricted to specific kinds of metrics or small problem sizes because of the large number of metric constraints in such problems. In this paper, we provide an active set algorithm, Project and Forget, that uses Bregman projections, to solve metric constrained problems with many (possibly exponentially) inequality constraints. We provide a theoretical analysis of Project and Forget and prove that our algorithm converges to the global optimal solution and that the L_2 distance of the current iterate to the optimal solution decays asymptotically at an exponential rate. We demonstrate that using our method we can solve large problem instances of three types of metric constrained problems: general weight correlation clustering, metric nearness, and metric learning; in each case, out-performing the state of the art methods with respect to CPU times and problem sizes.

In representation learning, uniformity refers to the uniform feature distribution in the latent space (i.e., unit hypersphere). Previous work has shown that improving uniformity contributes to the learning of under-represented classes. However, most of the previous work focused on classification; the representation space of imbalanced regression remains unexplored. Classification-based methods are not suitable for regression tasks because they cluster features into distinct groups without considering the continuous and ordered nature essential for regression. In a geometric aspect, we uniquely focus on ensuring uniformity in the latent space for imbalanced regression through two key losses: enveloping and homogeneity. The enveloping loss encourages the induced trace to uniformly occupy the surface of a hypersphere, while the homogeneity loss ensures smoothness, with representations evenly spaced at consistent intervals. Our method integrates these geometric principles into the data representations via a Surrogate-driven Representation Learning (SRL) framework. Experiments with real-world regression and operator learning tasks highlight the importance of uniformity in imbalanced regression and validate the efficacy of our geometry-based loss functions.

Do contemporary diffusion models preserve the class geometry of hyperspherical data? Standard diffusion models rely on isotropic Gaussian noise in the forward process, inherently favoring Euclidean spaces. However, many real-world problems involve non-Euclidean distributions, such as hyperspherical manifolds, where class-specific patterns are governed by angular geometry within hypercones. When modeled in Euclidean space, these angular subtleties are lost, leading to suboptimal generative performance. To address this limitation, we introduce HyperSphereDiff to align hyperspherical structures with directional noise, preserving class geometry and effectively capturing angular uncertainty. We demonstrate both theoretically and empirically that this approach aligns the generative process with the intrinsic geometry of hyperspherical data, resulting in more accurate and geometry-aware generative models. We evaluate our framework on four object datasets and two face datasets, showing that incorporating angular uncertainty better preserves the underlying hyperspherical manifold. Resources are available at: {https://github.com/IAB-IITJ/Harmonizing-Geometry-and-Uncertainty-Diffusion-with-Hyperspheres/}

Large vision-language models have significantly advanced GUI agents, enabling executable interaction across web, mobile, and desktop interfaces. Yet these gains largely rely on a forgiving region-tolerant paradigm, where many nearby pixels inside the same component remain valid. Precise geometric construction breaks this assumption: actions must land on points in continuous canvas space rather than tolerant regions. Because geometric primitives carry ontological dependencies, a local coordinate error can induce cascading topological failures that distort downstream objects and invalidate the final construction. We identify this regime as precision-sensitive GUI tasks, requiring point-level accuracy, geometry-aware verification, and robustness to dependency-driven error propagation. To benchmark it, we introduce PAGE Bench, with 4,906 problems and over 224K process-supervised, pixel-level GUI actions. We further propose PAGER, a topology-aware agent that decomposes construction into dependency-structured planning and pixel-level execution. Pixel-grounded supervised tuning establishes executable action grammar, while precision-aligned reinforcement learning mitigates rollout-induced exposure bias through state-conditioned geometric feedback. Experiments reveal a pronounced Semantic-Execution Gap: general multimodal models can exceed 88% action type accuracy yet remain below 6% task success. PAGER closes this gap, delivering 4.1x higher task success than the strongest evaluated general baseline and raising step success rate from below 9% for GUI-specialized agents to over 62%, establishing a new state of the art for point-precise GUI control.

Proving geometric theorems constitutes a hallmark of visual reasoning combining both intuitive and logical skills. Therefore, automated theorem proving of Olympiad-level geometry problems is considered a notable milestone in human-level automated reasoning. The introduction of AlphaGeometry, a neuro-symbolic model trained with 100 million synthetic samples, marked a major breakthrough. It solved 25 of 30 International Mathematical Olympiad (IMO) problems whereas the reported baseline based on Wu's method solved only ten. In this note, we revisit the IMO-AG-30 Challenge introduced with AlphaGeometry, and find that Wu's method is surprisingly strong. Wu's method alone can solve 15 problems, and some of them are not solved by any of the other methods. This leads to two key findings: (i) Combining Wu's method with the classic synthetic methods of deductive databases and angle, ratio, and distance chasing solves 21 out of 30 methods by just using a CPU-only laptop with a time limit of 5 minutes per problem. Essentially, this classic method solves just 4 problems less than AlphaGeometry and establishes the first fully symbolic baseline strong enough to rival the performance of an IMO silver medalist. (ii) Wu's method even solves 2 of the 5 problems that AlphaGeometry failed to solve. Thus, by combining AlphaGeometry with Wu's method we set a new state-of-the-art for automated theorem proving on IMO-AG-30, solving 27 out of 30 problems, the first AI method which outperforms an IMO gold medalist.

We introduce `dualGNN', an autoregressive message-passing GNN for sampling fine, regular triangulations (FRTs) of convex polytopes. dualGNN operates on a generalization of the dual graph of a triangulation, with edges labeled by `signed circuits' -- combinatorial invariants from oriented matroid theory which we show are both necessary and sufficient for exposing regularity. The model is independent of the number of points in the polytope and invariant under the polytope's orientation-preserving symmetries (SL(d,Z) ltimes Z^d). When implemented with a certain masking procedure, one can also guarantee that every rollout produces a fine triangulation (in 2D). On unseen polygons with N_pts leq 40, dualGNN is the most uniform FRT sampler we tested, and even a model trained on a single polygon generalizes well to other polygons. The model is small (sim92k parameters), trains in sim7.5 hours on a single consumer GPU, and runs without modification on an M1 MacBook Pro. We apply dualGNN to string theory, uniformly sampling Calabi-Yau threefolds at h^{1,1}=86 and consistent with uniformity at h^{1,1}=128. This is an order of magnitude beyond previous learned methods with a model sim1000times smaller. Code, training scripts, and pretrained models are available at https://github.com/natemacfadden/dualGNN .

Spatial consistency is a fundamental property of the visual world and a key requirement for models that aim to understand physical reality. Despite recent advances, multimodal large language models (MLLMs) often struggle to reason about 3D geometry across multiple views. Rather than asking models to describe scene attributes, we introduce a more challenging task: given two views of the same scene, identify the object that violates 3D motion consistency. We propose a simple and scalable method for generating realistic, spatially inconsistent image pairs from multi-view scenes, enabling systematic evaluation of this capability. Our results show that state-of-the-art MLLMs significantly underperform human observers and exhibit substantial variability across different scene attributes, revealing a fragile and incomplete understanding of 3D structure. We hope our findings underscore the need for approaches that develop a more deeply grounded understanding of the physical world.

Plane Geometry Diagram Synthesis has been a crucial task in computer graphics, with applications ranging from educational tools to AI-driven mathematical reasoning. Traditionally, we rely on manual tools (e.g., Matplotlib and GeoGebra) to generate precise diagrams, but this usually requires huge, complicated calculations. Recently, researchers start to work on model-based methods (e.g., Stable Diffusion and GPT5) to automatically generate diagrams, saving operational cost but usually suffering from limited realism and insufficient accuracy. In this paper, we propose a novel framework GeoSDF, to automatically generate diagrams efficiently and accurately with Signed Distance Field (SDF). Specifically, we first represent geometric elements (e.g., points, segments, and circles) in the SDF, then construct a series of constraint functions to represent geometric relationships. Next, we optimize those constructed constraint functions to get an optimized field of both elements and constraints. Finally, by rendering the optimized field, we can obtain the synthesized diagram. In our GeoSDF, we define a symbolic language to represent geometric elements and constraints, and our synthesized geometry diagrams can be self-verified in the SDF, ensuring both mathematical accuracy and visual plausibility. In experiments, through both qualitative and quantitative analysis, GeoSDF synthesized both normal high-school level and IMO-level geometry diagrams. We achieve 88.67\% synthesis accuracy by human evaluation in the IMO problem set. Furthermore, we obtain a very high accuracy of solving geometry problems (over 95\% while the current SOTA accuracy is around 75%) by leveraging our self-verification property. All of these demonstrate the advantage of GeoSDF, paving the way for more sophisticated, accurate, and flexible generation of geometric diagrams for a wide array of applications.

Diffusion models for single image novel view synthesis (NVS) can generate highly realistic and plausible images, but they are limited in the geometric consistency to the given relative poses. The generated images often show significant errors with respect to the epipolar constraints that should be fulfilled, as given by the target pose. In this paper we address this issue by proposing a methodology to improve the geometric correctness of images generated by a diffusion model for single image NVS. We formulate a loss function based on image matching and epipolar constraints, and optimize the starting noise in a diffusion sampling process such that the generated image should both be a realistic image and fulfill geometric constraints derived from the given target pose. Our method does not require training data or fine-tuning of the diffusion models, and we show that we can apply it to multiple state-of-the-art models for single image NVS. The method is evaluated on the MegaScenes dataset and we show that geometric consistency is improved compared to the baseline models while retaining the quality of the generated images.

Geometry problem solving is a well-recognized testbed for evaluating the high-level multi-modal reasoning capability of deep models. In most existing works, two main geometry problems: calculation and proving, are usually treated as two specific tasks, hindering a deep model to unify its reasoning capability on multiple math tasks. However, in essence, these two tasks have similar problem representations and overlapped math knowledge which can improve the understanding and reasoning ability of a deep model on both two tasks. Therefore, we construct a large-scale Unified Geometry problem benchmark, UniGeo, which contains 4,998 calculation problems and 9,543 proving problems. Each proving problem is annotated with a multi-step proof with reasons and mathematical expressions. The proof can be easily reformulated as a proving sequence that shares the same formats with the annotated program sequence for calculation problems. Naturally, we also present a unified multi-task Geometric Transformer framework, Geoformer, to tackle calculation and proving problems simultaneously in the form of sequence generation, which finally shows the reasoning ability can be improved on both two tasks by unifying formulation. Furthermore, we propose a Mathematical Expression Pretraining (MEP) method that aims to predict the mathematical expressions in the problem solution, thus improving the Geoformer model. Experiments on the UniGeo demonstrate that our proposed Geoformer obtains state-of-the-art performance by outperforming task-specific model NGS with over 5.6% and 3.2% accuracies on calculation and proving problems, respectively.

Score distillation sampling (SDS), the methodology in which the score from pretrained 2D diffusion models is distilled into 3D representation, has recently brought significant advancements in text-to-3D generation task. However, this approach is still confronted with critical geometric inconsistency problems such as the Janus problem. Starting from a hypothesis that such inconsistency problems may be induced by multiview inconsistencies between 2D scores predicted from various viewpoints, we introduce GSD, a simple and general plug-and-play framework for incorporating 3D consistency and therefore geometry awareness into the SDS process. Our methodology is composed of three components: 3D consistent noising, designed to produce 3D consistent noise maps that perfectly follow the standard Gaussian distribution, geometry-based gradient warping for identifying correspondences between predicted gradients of different viewpoints, and novel gradient consistency loss to optimize the scene geometry toward producing more consistent gradients. We demonstrate that our method significantly improves performance, successfully addressing the geometric inconsistency problems in text-to-3D generation task with minimal computation cost and being compatible with existing score distillation-based models. Our project page is available at https://ku-cvlab.github.io/GSD/.

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

In this paper we show that visual diffusion models can serve as effective geometric solvers: they can directly reason about geometric problems by working in pixel space. We first demonstrate this on the Inscribed Square Problem, a long-standing problem in geometry that asks whether every Jordan curve contains four points forming a square. We then extend the approach to two other well-known hard geometric problems: the Steiner Tree Problem and the Simple Polygon Problem. Our method treats each problem instance as an image and trains a standard visual diffusion model that transforms Gaussian noise into an image representing a valid approximate solution that closely matches the exact one. The model learns to transform noisy geometric structures into correct configurations, effectively recasting geometric reasoning as image generation. Unlike prior work that necessitates specialized architectures and domain-specific adaptations when applying diffusion to parametric geometric representations, we employ a standard visual diffusion model that operates on the visual representation of the problem. This simplicity highlights a surprising bridge between generative modeling and geometric problem solving. Beyond the specific problems studied here, our results point toward a broader paradigm: operating in image space provides a general and practical framework for approximating notoriously hard problems, and opens the door to tackling a far wider class of challenging geometric tasks.

Learning a 4D scene representation from a single monocular video that supports dynamic novel-view synthesis while maintaining faithful geometry over time remains challenging. Dynamic Gaussian Splatting achieves strong rendering performance through photometric optimization, yet does not explicitly enforce multi-view geometric consistency. In contrast, 3D foundation models recover coherent scene geometry and camera motion, but their point-based outputs are not designed for photorealistic rendering. We propose Ground4D, a geometry-grounded framework built on two stages. First, we perform geometry initialization via 3D foundation models, leveraging VGGT in a training-free manner to reconstruct multi-view-consistent 3D geometry and camera poses from monocular video. The recovered geometry provides a structured and reliable initialization for dynamic Gaussian representations. Second, we conduct geometry-consistency-aware refinement via dynamic Gaussian Splatting, optimizing the representation through differentiable rendering while maintaining multi-view geometric consistency across both observed and synthesized viewpoints. Furthermore, Ground4D inherently models the continuous 4D dynamics of the scene, naturally supporting rendering at arbitrary timestamps. By integrating foundation-level geometric priors into dynamic Gaussian optimization, Ground4D achieves stronger reconstruction fidelity and rendering performance, underscoring the role of geometry-grounded constraints in robust 4D scene modeling.

Predictive world models that simulate future observations under explicit camera control are fundamental to interactive AI. Despite rapid advances, current systems lack spatial persistence: they fail to maintain stable scene structures over long trajectories, frequently hallucinating details when cameras revisit previously observed locations. We identify that this geometric drift stems from reliance on screen-space positional embeddings, which conflict with the projective geometry required for 3D consistency. We introduce ViewRope, a geometry-aware encoding that injects camera-ray directions directly into video transformer self-attention layers. By parameterizing attention with relative ray geometry rather than pixel locality, ViewRope provides a model-native inductive bias for retrieving 3D-consistent content across temporal gaps. We further propose Geometry-Aware Frame-Sparse Attention, which exploits these geometric cues to selectively attend to relevant historical frames, improving efficiency without sacrificing memory consistency. We also present ViewBench, a diagnostic suite measuring loop-closure fidelity and geometric drift. Our results demonstrate that ViewRope substantially improves long-term consistency while reducing computational costs.

Previous works that leverage video models for image-to-3D scene generation often suffer from geometric distortions and blurry content. Using video generation models to implicitly maintain geometric consistency according to a single-frame input is ineffective. In this paper, we present a two-stage method, named GeoWorld, that renovates the image-to-3D scene generation pipeline by providing full-frame geometry features. The first-stage video generation model, followed by a multi-view geometry model, produces full-frame geometry features, which are then used as a mental draft of geometric conditions to aid the second-stage video-generation model. A geometric loss is proposed to impose real-world geometric constraints, and a geometry adaptation module is introduced to ensure the effective utilization of geometry features. Thanks to full-frame geometric modeling, the two smaller video models in our two-stage method can generate higher-fidelity 3D scenes than SOTA methods, while being even faster, e.g. 7.5times faster than Hunyuan-Voyager. Project page: https://peaes.github.io/GeoWorld.

A significant gap exists between theory and practice in deep learning. Generalization and approximation error bounds are often derived for simplified models or are too loose to be informative. Many rely on the manifold hypothesis and on geometric regularity such as intrinsic dimension, curvature, and reach. Progress requires insight into data-manifold geometry and suitable benchmarks, yet existing options are polarized: analytic manifolds with known geometry but limited applicability, or real-world datasets where geometry is only coarsely estimable. We introduce a benchmarking framework for studying data geometry. We repurpose and extend dSprites and COIL-20 with additional transformation dimensions and dense, axis-aligned sampling, and pair them with finite-difference estimators that recover curvature, reach, and volume at near-ground-truth accuracy in a regime where general-purpose estimators are unreliable or difficult to deploy. The framework is intended as a controlled testbed, useful as a calibration environment for geometric estimators and a sandbox for probing theoretical assumptions. To illustrate its use, we present two application studies, namely assessing the scaling behavior of the bounds of Genovese et al. and Fefferman et al., and tracking the layer-wise geometry of a β-VAE, highlighting the behavior of current bounds and the value of controlled benchmarks for guiding and validating future theory. A reference implementation is available at https://github.com/koulakis/manifold-microscope.

Generalized visual grounding tasks, including Generalized Referring Expression Comprehension (GREC) and Segmentation (GRES), extend the classical visual grounding paradigm by accommodating multi-target and non-target scenarios. Specifically, GREC focuses on accurately identifying all referential objects at the coarse bounding box level, while GRES aims for achieve fine-grained pixel-level perception. However, existing approaches typically treat these tasks independently, overlooking the benefits of jointly training GREC and GRES to ensure consistent multi-granularity predictions and streamline the overall process. Moreover, current methods often treat GRES as a semantic segmentation task, neglecting the crucial role of instance-aware capabilities and the necessity of ensuring consistent predictions between instance-level boxes and masks. To address these limitations, we propose InstanceVG, a multi-task generalized visual grounding framework equipped with instance-aware capabilities, which leverages instance queries to unify the joint and consistency predictions of instance-level boxes and masks. To the best of our knowledge, InstanceVG is the first framework to simultaneously tackle both GREC and GRES while incorporating instance-aware capabilities into generalized visual grounding. To instantiate the framework, we assign each instance query a prior reference point, which also serves as an additional basis for target matching. This design facilitates consistent predictions of points, boxes, and masks for the same instance. Extensive experiments obtained on ten datasets across four tasks demonstrate that InstanceVG achieves state-of-the-art performance, significantly surpassing the existing methods in various evaluation metrics. The code and model will be publicly available at https://github.com/Dmmm1997/InstanceVG.

Generalization, the ability to perform well beyond the training context, is a hallmark of biological and artificial intelligence, yet anticipating unseen failures remains a central challenge. Conventional approaches often take a ``bottom-up'' mechanistic route by reverse-engineering interpretable features or circuits to build explanatory models. While insightful, these methods often struggle to provide the high-level, predictive signals for anticipating failure in real-world deployment. Here, we propose using a ``top-down'' approach to studying generalization failures inspired by medical biomarkers: identifying system-level measurements that serve as robust indicators of a model's future performance. Rather than mapping out detailed internal mechanisms, we systematically design and test network markers to probe structure, function links, identify prognostic indicators, and validate predictions in real-world settings. In image classification, we find that task-relevant geometric properties of in-distribution (ID) object manifolds consistently forecast poor out-of-distribution (OOD) generalization. In particular, reductions in two geometric measures, effective manifold dimensionality and utility, predict weaker OOD performance across diverse architectures, optimizers, and datasets. We apply this finding to transfer learning with ImageNet-pretrained models. We consistently find that the same geometric patterns predict OOD transfer performance more reliably than ID accuracy. This work demonstrates that representational geometry can expose hidden vulnerabilities, offering more robust guidance for model selection and AI interpretability.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

Continual post-training aims to extend large language models (LLMs) with new knowledge, skills, and behaviors, yet it remains unclear when sequential updates enable capability transfer and when they cause catastrophic forgetting. Existing methods mitigate forgetting through sequential fine-tuning, replay, regularization, or model merging, but offer limited criteria for determining when incorporating new updates is beneficial or harmful. In this work, we study LLM continual post-training through three questions: What drives forgetting? When do sequentially acquired capabilities transfer or interfere? How can compatibility be used to control update integration? We address these questions through task geometry: we represent each post-training task by its parameter update and study the covariance geometry induced by the update. Our central finding is that: forgetting can be considered as a state-relative update-integration failure, it arises when the covariance geometries induced by tasks misalign with the geometry of the evolving model state. Sequential updates transfer when they remain compatible with the model state shaped by previous updates, and interfere when state-relative geometry conflict becomes high. Motivated by this finding, we propose Geometry-Conflict Wasserstein Merging (GCWM), a data-free update-integration method that constructs a shared Wasserstein metric via Gaussian Wasserstein barycenters and uses geometry conflict to gate geometry-aware correction. Across Qwen3 0.6B--14B on domain-continual and capability-continual settings, GCWM consistently outperforms data-free baselines, improving retention and final performance without replay data. These results identify geometry conflict as both an explanatory signal for forgetting and a practical control signal for LLM continual post-training.

Computer-aided design (CAD) is the digital construction of 2D and 3D objects, and is central to a wide range of engineering and manufacturing applications like automobile and aviation. Despite its importance, CAD modeling remains largely a time-intensive, manual task. Recent works have attempted to automate this process with small transformer-based models and handcrafted CAD sequence representations. However, there has been little effort to leverage the potential of large language models (LLMs) for sequential CAD design. In this work, we introduce a new large-scale dataset of more than 170k CAD models annotated with high-quality, human-like descriptions generated with our pipeline based on GPT-4.1. Using this dataset, we fine-tune powerful code-LLMs to generate CAD sequences represented in a JSON-based format from natural language descriptions, demonstrating the viability and effectiveness of this approach for text-conditioned CAD generation. Because simple metrics often fail to reflect the quality of generated objects, we introduce geometric and topological metrics based on sphericity, mean curvature, and Euler characteristic to provide richer structural insights. Our experiments and ablation studies on both synthetic and human-annotated data demonstrate that CADmium is able to automate CAD design, drastically speeding up the design of new objects. The dataset, code, and fine-tuned models are available online.

Geometric foundation models, such as the Visual Geometry Grounded Transformer (VGGT), provide strong 3D priors from unposed images. However, such models operate purely in a feed-forward, deterministic regime, \ie~they cannot generate plausible geometry beyond what the input views directly support. Generative models for 3D scenes, on the other hand, must rely on strong geometric priors to produce coherent outputs from sparse inputs. We bridge these two paradigms by performing flow matching directly in VGGT's latent space, leveraging its learned 3D priors without committing to any explicit downstream representation such as Gaussians, meshes, or video-VAE latents. This requires respecting the latent geometry: VGGT tokens occupy a product of high-dimensional hyperspheres on which standard Euclidean flow matching fails. We address this with a Riemannian Flow Matching framework defined on a product manifold of four hyperspheres, aligned with VGGT's multi-scale encoder, which keeps generated tokens on the valid data manifold required by the frozen decoding heads. On RealEstate10K, ScanNet++ and ETH3D, our method achieves strong performance against recent scene generation baselines in both per-view appearance and aggregated 3D geometry, establishing latent-space flow matching on geometric foundation models as a viable paradigm for 3D generation. The project page can be found https://lisaweijler.github.io/geometry-grounded-rfm/{here}.

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.

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.

Reliable deployment of language models requires two capabilities that appear distinct but share a common geometric foundation: predicting whether a model will accept targeted behavioral control, and detecting when its internal structure degrades. We show that geometric stability, the consistency of a representation's pairwise distance structure, addresses both. Supervised Shesha variants that measure task-aligned geometric stability predict linear steerability with near-perfect accuracy (ρ= 0.89-0.97) across 35-69 embedding models and three NLP tasks, capturing unique variance beyond class separability (partial ρ= 0.62-0.76). A critical dissociation emerges: unsupervised stability fails entirely for steering on real-world tasks (ρapprox 0.10), revealing that task alignment is essential for controllability prediction. However, unsupervised stability excels at drift detection, measuring nearly 2times greater geometric change than CKA during post-training alignment (up to 5.23times in Llama) while providing earlier warning in 73\% of models and maintaining a 6times lower false alarm rate than Procrustes. Together, supervised and unsupervised stability form complementary diagnostics for the LLM deployment lifecycle: one for pre-deployment controllability assessment, the other for post-deployment monitoring.

Industrial visual sim-to-real is often described as transferring from synthetic images to real images, but industrial deployment usually involves a broader mismatch between available evidence and required decisions. A system may be built from CAD renderings, simulated RGB-D observations, normal reference images, synthetic defects, pretrained feature spaces, or language prompts, yet deployed under different sensors, lighting, materials, fixtures, calibration, production variation, and rare defect modes. This review reframes industrial visual sim-to-real as a domain-gap problem organized by prior availability. We distinguish CAD-available settings, where explicit object geometry can support rendering, calibration, pose estimation, segmentation, and test-time geometric verification; CAD-unavailable settings, where geometry is replaced by normal-reference appearance, feature distributions, teacher-student residuals, synthetic anomaly assumptions, foundation features, or vision-language priors; and boundary-prior settings, where approximate models, templates, reference views, or semantic correspondences preserve only part of the CAD role. This framing connects CAD-based detection and 6D pose-estimation literature with industrial anomaly and surface-inspection literature that is usually reviewed separately. To make the taxonomy concrete, we use empirical anchors on T-LESS/BOP, MVTec AD, and VisA. The anchors show that CAD render count alone does not close transfer; source-distribution design, detector capacity, and small real calibration can matter more. They also show that CAD at test time creates a distinct verification channel through mask, pose, and depth consistency, whereas CAD-unavailable inspection relies on calibrated normality and feature deviation. The review therefore argues against a single cross-task leaderboard and instead asks what prior grounds the deployment decision.

The Distance Geometry Problem (DGP) seeks to find positions for a set of points in geometric space when some distances between pairs of these points are known. The so-called discretization assumptions allow to discretize the search space of DGP instances. In this paper, we study the Discretizable Molecular Distance Geometry Problem whose feasible solutions provide a discretization scheme for the DGP. We propose the first constraint programming formulations as well as a set of checks for proving infeasibility, domain reduction techniques, symmetry breaking constraints and valid inequalities. Our computational results indicate that our formulations outperform the state-of-the-art integer programming formulations, both for feasible and infeasible instances.

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

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.

Geometry problem solving (GPS) represents a critical frontier in artificial intelligence, with profound applications in education, computer-aided design, and computational graphics. Despite its significance, automating GPS remains challenging due to the dual demands of spatial understanding and rigorous logical reasoning. Recent advances in large models have enabled notable breakthroughs, particularly for SAT-level problems, yet the field remains fragmented across methodologies, benchmarks, and evaluation frameworks. This survey systematically synthesizes GPS advancements through three core dimensions: (1) benchmark construction, (2) textual and diagrammatic parsing, and (3) reasoning paradigms. We further propose a unified analytical paradigm, assess current limitations, and identify emerging opportunities to guide future research toward human-level geometric reasoning, including automated benchmark generation and interpretable neuro-symbolic integration.

Spatial reasoning over three-dimensional scenes is a core capability for embodied intelligence, yet continuous model improvement remains bottlenecked by the cost of geometric annotation. The self-evolving paradigm offers a promising path, but its reliance on model consensus to construct pseudo-labels causes training to reinforce rather than correct the model's own geometric errors. We identify a property unique to 3D spatial reasoning that circumvents this limitation: ground truth is a deterministic consequence of the underlying geometry, computable exactly from point clouds and camera poses without any model involvement. Building on this insight, we present SpatialEvo, a self-evolving framework for 3D spatial reasoning, centered on the Deterministic Geometric Environment (DGE). The DGE formalizes 16 spatial reasoning task categories under explicit geometric validation rules and converts unannotated 3D scenes into zero-noise interactive oracles, replacing model consensus with objective physical feedback. A single shared-parameter policy co-evolves across questioner and solver roles under DGE constraints: the questioner generates physically valid spatial questions grounded in scene observations, while the solver derives precise answers against DGE-verified ground truth. A task-adaptive scheduler endogenously concentrates training on the model's weakest categories, producing a dynamic curriculum without manual design. Experiments across nine benchmarks demonstrate that SpatialEvo achieves the highest average score at both 3B and 7B scales, with consistent gains on spatial reasoning benchmarks and no degradation on general visual understanding.

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.

Generating executable CAD programs from images requires alignment between visual geometry and symbolic program representations, a capability that current methods fail to learn reliably as design complexity increases. Existing fine-tuning approaches rely on either limited supervised datasets or expensive post-training pipelines, resulting in brittle systems that restrict progress in generative CAD design. We argue that the primary bottleneck lies not in model or algorithmic capacity, but in the scarcity of diverse training examples that align visual geometry with program syntax. This limitation is especially acute because the collection of diverse and verified engineering datasets is both expensive and difficult to scale, constraining the development of robust generative CAD models. We introduce Geometric Inference Feedback Tuning (GIFT), a data augmentation framework that leverages geometric feedback to turn test-time compute into a bootstrapped set of high-quality training samples. GIFT combines two mechanisms: Soft-Rejection Sampling (GIFT-REJECT), which retains diverse high-fidelity programs beyond exact ground-truth matches, and Failure-Driven Augmentation (GIFT-FAIL), which converts near-miss predictions into synthetic training examples that improve robustness on challenging geometries. By amortizing inference-time search into the model parameters, GIFT captures the benefits of test-time scaling while reducing inference compute by 80%. It improves mean IoU by 12% over a strong supervised baseline and remains competitive with more complex multimodal systems, without requiring additional human annotation or specialized architectures.

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

Empirically validating new 3D-printing related algorithms and implementations requires testing data representative of inputs encountered in the wild. An ideal benchmarking dataset should not only draw from the same distribution of shapes people print in terms of class (e.g., toys, mechanisms, jewelry), representation type (e.g., triangle soup meshes) and complexity (e.g., number of facets), but should also capture problems and artifacts endemic to 3D printing models (e.g., self-intersections, non-manifoldness). We observe that the contextual and geometric characteristics of 3D printing models differ significantly from those used for computer graphics applications, not to mention standard models (e.g., Stanford bunny, Armadillo, Fertility). We present a new dataset of 10,000 models collected from an online 3D printing model-sharing database. Via analysis of both geometric (e.g., triangle aspect ratios, manifoldness) and contextual (e.g., licenses, tags, classes) characteristics, we demonstrate that this dataset represents a more concise summary of real-world models used for 3D printing compared to existing datasets. To facilitate future research endeavors, we also present an online query interface to select subsets of the dataset according to project-specific characteristics. The complete dataset and per-model statistical data are freely available to the public.

We present a computational framework that transforms single images into 3D physical objects. The visual geometry of a physical object in an image is determined by three orthogonal attributes: mechanical properties, external forces, and rest-shape geometry. Existing single-view 3D reconstruction methods often overlook this underlying composition, presuming rigidity or neglecting external forces. Consequently, the reconstructed objects fail to withstand real-world physical forces, resulting in instability or undesirable deformation -- diverging from their intended designs as depicted in the image. Our optimization framework addresses this by embedding physical compatibility into the reconstruction process. We explicitly decompose the three physical attributes and link them through static equilibrium, which serves as a hard constraint, ensuring that the optimized physical shapes exhibit desired physical behaviors. Evaluations on a dataset collected from Objaverse demonstrate that our framework consistently enhances the physical realism of 3D models over existing methods. The utility of our framework extends to practical applications in dynamic simulations and 3D printing, where adherence to physical compatibility is paramount.

Mathematical geometric reasoning is essential for scientific discovery and educational development, requiring precise logic and rigorous formal verification. While recent advances in Multimodal Large Language Models (MLLMs) have improved reasoning tasks, existing models typically struggle with formal geometric reasoning, particularly when dynamically constructing and verifying auxiliary geometric elements. To address these challenges, we introduce Geoint-R1, a multimodal reasoning framework designed to generate formally verifiable geometric solutions from textual descriptions and visual diagrams. Geoint-R1 uniquely integrates auxiliary elements construction, formal reasoning represented via Lean4, and interactive visualization. To systematically evaluate and advance formal geometric reasoning, we propose the Geoint benchmark, comprising 1,885 rigorously annotated geometry problems across diverse topics such as plane, spatial, and solid geometry. Each problem includes structured textual annotations, precise Lean4 code for auxiliary constructions, and detailed solution steps verified by experts. Extensive experiments demonstrate that Geoint-R1 significantly surpasses existing multimodal and math-specific reasoning models, particularly on challenging problems requiring explicit auxiliary element constructions.

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.

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

Mathematical geometric problem solving (GPS) often requires effective integration of multimodal information and verifiable logical coherence. Despite the fast development of large language models in general problem solving, it remains unresolved regarding with both methodology and benchmarks, especially given the fact that exiting synthetic GPS benchmarks are often not self-verified and contain noise and self-contradicted information due to the illusion of LLMs. In this paper, we propose a scalable data engine called TrustGeoGen for problem generation, with formal verification to provide a principled benchmark, which we believe lays the foundation for the further development of methods for GPS. The engine synthesizes geometric data through four key innovations: 1) multimodal-aligned generation of diagrams, textual descriptions, and stepwise solutions; 2) formal verification ensuring rule-compliant reasoning paths; 3) a bootstrapping mechanism enabling complexity escalation via recursive state generation and 4) our devised GeoExplore series algorithms simultaneously produce multi-solution variants and self-reflective backtracking traces. By formal logical verification, TrustGeoGen produces GeoTrust-200K dataset with guaranteed modality integrity, along with GeoTrust-test testset. Experiments reveal the state-of-the-art models achieve only 49.17\% accuracy on GeoTrust-test, demonstrating its evaluation stringency. Crucially, models trained on GeoTrust achieve OOD generalization on GeoQA, significantly reducing logical inconsistencies relative to pseudo-label annotated by OpenAI-o1. Our code is available at https://github.com/Alpha-Innovator/TrustGeoGen

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.

From an image of a person, we can easily infer the natural 3D pose and shape of the person even if ambiguity exists. This is because we have a mental model that allows us to imagine a person's appearance at different viewing directions from a given image and utilize the consistency between them for inference. However, existing human mesh recovery methods only consider the direction in which the image was taken due to their structural limitations. Hence, we propose "Implicit 3D Human Mesh Recovery (ImpHMR)" that can implicitly imagine a person in 3D space at the feature-level via Neural Feature Fields. In ImpHMR, feature fields are generated by CNN-based image encoder for a given image. Then, the 2D feature map is volume-rendered from the feature field for a given viewing direction, and the pose and shape parameters are regressed from the feature. To utilize consistency with pose and shape from unseen-view, if there are 3D labels, the model predicts results including the silhouette from an arbitrary direction and makes it equal to the rotated ground-truth. In the case of only 2D labels, we perform self-supervised learning through the constraint that the pose and shape parameters inferred from different directions should be the same. Extensive evaluations show the efficacy of the proposed method.

In this paper, we present a deep learning-based framework for solving geometric construction problems through visual reasoning, which is useful for automated geometry theorem proving. Constructible problems in geometry often ask for the sequence of straightedge-and-compass constructions to construct a given goal given some initial setup. Our EuclidNet framework leverages the neural network architecture Mask R-CNN to extract the visual features from the initial setup and goal configuration with extra points of intersection, and then generate possible construction steps as intermediary data models that are used as feedback in the training process for further refinement of the construction step sequence. This process is repeated recursively until either a solution is found, in which case we backtrack the path for a step-by-step construction guide, or the problem is identified as unsolvable. Our EuclidNet framework is validated on complex Japanese Sangaku geometry problems, demonstrating its capacity to leverage backtracking for deep visual reasoning of challenging problems.

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

Real-world phenomena do not generate arbitrary variability: their signals concentrate on compact, low-variability subsets of functional space, enabling rapid generalization from few examples. A small child can recognize a dog after extremely limited exposure because the perceptual manifold of "dog" is compact, structured, and low-dimensional. We formalize this principle through a deterministic functional-topological framework in which the set of valid realizations produced by a physical process forms a compact subset of a Banach space, endowed with stable invariants, a finite Hausdorff radius, and an induced continuous perceptual functional. This geometry provides explicit limits on knowledge, conditions for identifiability, and guarantees for generalization from sparse evidence -- properties fundamental to both natural and artificial intelligence. Across electromechanical, electrochemical, and physiological domains, we show that real-world processes consistently generate compact perceptual manifolds with the same geometric characteristics. Their boundaries can be discovered in a fully self-supervised manner as the empirical radius saturates with increasing sampling, even when the governing equations are unknown. These results demonstrate that deterministic functional topology offers a unified mathematical foundation for perception, representation, and world-model construction. It provides a geometric explanation for why biological learners and self-supervised AI systems can generalize from few observations, and establishes compact perceptual manifolds as a fundamental building block for future AI architectures. Finally, this work unifies biological perception and modern self-supervised models under a single geometric principle: both derive their generalization ability from the compactness and invariants of real-world perceptual manifolds.

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 propose MoGe-2, an advanced open-domain geometry estimation model that recovers a metric scale 3D point map of a scene from a single image. Our method builds upon the recent monocular geometry estimation approach, MoGe, which predicts affine-invariant point maps with unknown scales. We explore effective strategies to extend MoGe for metric geometry prediction without compromising the relative geometry accuracy provided by the affine-invariant point representation. Additionally, we discover that noise and errors in real data diminish fine-grained detail in the predicted geometry. We address this by developing a unified data refinement approach that filters and completes real data from different sources using sharp synthetic labels, significantly enhancing the granularity of the reconstructed geometry while maintaining the overall accuracy. We train our model on a large corpus of mixed datasets and conducted comprehensive evaluations, demonstrating its superior performance in achieving accurate relative geometry, precise metric scale, and fine-grained detail recovery -- capabilities that no previous methods have simultaneously achieved.

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.

With the rise of deep neural networks, especially in safety-critical applications, robustness and interpretability are crucial to ensure their trustworthiness. Recent advances in 3D-aware classifiers that map image features to volumetric representation of objects, rather than relying solely on 2D appearance, have greatly improved robustness on out-of-distribution (OOD) data. Such classifiers have not yet been studied from the perspective of interpretability. Meanwhile, current concept-based XAI methods often neglect OOD robustness. We aim to address both aspects with CAVE - Concept Aware Volumes for Explanations - a new direction that unifies interpretability and robustness in image classification. We design CAVE as a robust and inherently interpretable classifier that learns sparse concepts from 3D object representation. We further propose 3D Consistency (3D-C), a metric to measure spatial consistency of concepts. Unlike existing metrics that rely on human-annotated parts on images, 3D-C leverages ground-truth object meshes as a common surface to project and compare explanations across concept-based methods. CAVE achieves competitive classification performance while discovering consistent and meaningful concepts across images in various OOD settings. Code available at https://github.com/phamleyennhi/CAVE.

Large Language Models exhibit paradoxical fragility in fundamental arithmetic, implying a disconnect between internal computation and discrete output. By analyzing the residual stream geometry during multi-operand addition, we identify the Iso-Raw-Sum Trajectory (IRST), a geometric structure where representations are anchored by semantic digits and modulated by continuous carry fibers. We propose the Noisy Quantization Model to explain this geometry, framing arithmetic errors as Geometric Slippages caused by internal neural noise pushing a continuous, latent Carry Potential across quantization thresholds. This geometric framework further elucidates Probe Versatility, explaining how lightweight probes can disentangle coexisting latent signals (such as ground truth versus hallucination) from a single activation vector. Finally, we validate these insights through a geometric consistency check method that effectively detects and corrects these quantization failures during inference. Our code is available at https://github.com/RL-MIND/Shape-of-Addition.

Recent feed-forward models have significantly advanced geometry perception for inferring dense 3D structure from sensor observations. However, its essential capabilities remain fragmented across multiple incompatible paradigms, including online perception, offline reconstruction, multi-modal integration, long-horizon scalability, and metric-scale estimation. We present UniT, a unified model built upon a novel Group Autoregressive Transformer, which reformulates these seemingly disparate capabilities within a single framework. The key idea is to treat groups of sensor observations as the basic autoregressive units and predict the corresponding point maps in an anchor-free and scale-adaptive manner. More specifically, diverse view configurations in both online and offline settings are naturally unified within a single group autoregression process. By varying the group size, online mode operates over multiple autoregressive steps with single-frame groups, whereas offline mode aggregates a multi-frame group in a single forward pass. Meanwhile, a queue-style KV caching mechanism ensures bounded autoregressive memory over long horizons. This is enabled by reducing long-range dependencies on early frames through anchor-free relational modeling, thereby allowing outdated memory to be discarded on the fly. To improve metric-scale generalization across scenes, a scale-adaptive geometry loss is further introduced within this framework. It couples relative geometric constraints with a partial absolute scale term, implicitly regularizing global scale and inducing a progressive transition from scale-invariant geometry to metric-scale solutions. Together with a dedicated modal attention module for integrating auxiliary modalities, UniT achieves state-of-the-art performance in unified geometry perception, as validated on ten benchmarks spanning seven representative tasks.

Image-based 3D generation has vast applications in robotics and gaming, where high-quality, diverse outputs and consistent 3D representations are crucial. However, existing methods have limitations: 3D diffusion models are limited by dataset scarcity and the absence of strong pre-trained priors, while 2D diffusion-based approaches struggle with geometric consistency. We propose a method that leverages 2D diffusion models' implicit 3D reasoning ability while ensuring 3D consistency via Gaussian-splatting-based geometric distillation. Specifically, the proposed Gaussian Splatting Decoder enforces 3D consistency by transforming SV3D latent outputs into an explicit 3D representation. Unlike SV3D, which only relies on implicit 2D representations for video generation, Gaussian Splatting explicitly encodes spatial and appearance attributes, enabling multi-view consistency through geometric constraints. These constraints correct view inconsistencies, ensuring robust geometric consistency. As a result, our approach simultaneously generates high-quality, multi-view-consistent images and accurate 3D models, providing a scalable solution for single-image-based 3D generation and bridging the gap between 2D Diffusion diversity and 3D structural coherence. Experimental results demonstrate state-of-the-art multi-view consistency and strong generalization across diverse datasets. The code will be made publicly available upon acceptance.

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

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.

Modern monocular 3D reconstruction methods and vision-language models (VLMs) demonstrate impressive results on standard benchmarks, yet recent works cast doubt on their true understanding of geometric properties. We introduce GOQ, a comprehensive benchmark specifically designed to evaluate the geometric reasoning capabilities of vision and vision-language foundation models. GIQ comprises synthetic and real-world images and corresponding 3D meshes of diverse polyhedra covering varying levels of complexity and symmetry, from Platonic, Archimedean, Johnson, and Catalan solids to stellations and compound shapes. Through systematic experiments involving monocular 3D reconstruction, 3D symmetry detection, mental rotation tests, and zero-shot shape classification tasks, we reveal significant shortcomings in current models. State-of-the-art reconstruction algorithms trained on extensive 3D datasets struggle to reconstruct even basic geometric Platonic solids accurately. Next, although foundation models may be shown via linear and non-linear probing to capture specific 3D symmetry elements, they falter significantly in tasks requiring detailed geometric differentiation, such as mental rotation. Moreover, advanced vision-language assistants such as ChatGPT, Gemini and Claud exhibit remarkably low accuracy in interpreting basic shape properties such as face geometry, convexity, and compound structures of complex polyhedra. GIQ is publicly available at toomanymatts.github.io/giq-benchmark/, providing a structured platform to benchmark critical gaps in geometric intelligence and facilitate future progress in robust, geometry-aware representation learning.

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.

Even in controlled settings, understanding instance-wise geometries is a challenging task for a wide range of visual models. Although specialized systems exist, modern arts rely on expensive input formats (category labels, binary segmentation masks) and inference costs (a quadratic amount of forward passes). We mitigate these limitations by proposing InstaFormer, a network capable of holistic order prediction. That is, solely given an input RGB image, InstaFormer returns the full occlusion and depth orderings for all the instances in the scene in a single forward pass. At its core, InstaFormer relies on interactions between object queries and latent mask descriptors that semantically represent the same objects while carrying complementary information. We comprehensively benchmark and ablate our approach to highlight its effectiveness. Our code and models are open-source and available at this URL: https://github.com/SNU-VGILab/InstaOrder.

Mathematics olympiads are prestigious competitions, with problem proposing and solving highly honored. Building artificial intelligence that proposes and solves olympiads presents an unresolved challenge in automated theorem discovery and proving, especially in geometry for its combination of numerical and spatial elements. We introduce TongGeometry, a Euclidean geometry system supporting tree-search-based guided problem proposing and solving. The efficient geometry system establishes the most extensive repository of geometry theorems to date: within the same computational budget as the existing state-of-the-art, TongGeometry discovers 6.7 billion geometry theorems requiring auxiliary constructions, including 4.1 billion exhibiting geometric symmetry. Among them, 10 theorems were proposed to regional mathematical olympiads with 3 of TongGeometry's proposals selected in real competitions, earning spots in a national team qualifying exam or a top civil olympiad in China and the US. Guided by fine-tuned large language models, TongGeometry solved all International Mathematical Olympiad geometry in IMO-AG-30, outperforming gold medalists for the first time. It also surpasses the existing state-of-the-art across a broader spectrum of olympiad-level problems. The full capabilities of the system can be utilized on a consumer-grade machine, making the model more accessible and fostering widespread democratization of its use. By analogy, unlike existing systems that merely solve problems like students, TongGeometry acts like a geometry coach, discovering, presenting, and proving theorems.

Recent advancements in human avatar synthesis have utilized radiance fields to reconstruct photo-realistic animatable human avatars. However, both NeRFs-based and 3DGS-based methods struggle with maintaining 3D consistency and exhibit suboptimal detail reconstruction, especially with sparse inputs. To address this challenge, we propose CHASE, which introduces supervision from intrinsic 3D consistency across poses and 3D geometry contrastive learning, achieving performance comparable with sparse inputs to that with full inputs. Following previous work, we first integrate a skeleton-driven rigid deformation and a non-rigid cloth dynamics deformation to coordinate the movements of individual Gaussians during animation, reconstructing basic avatar with coarse 3D consistency. To improve 3D consistency under sparse inputs, we design Dynamic Avatar Adjustment(DAA) to adjust deformed Gaussians based on a selected similar pose/image from the dataset. Minimizing the difference between the image rendered by adjusted Gaussians and the image with the similar pose serves as an additional form of supervision for avatar. Furthermore, we propose a 3D geometry contrastive learning strategy to maintain the 3D global consistency of generated avatars. Though CHASE is designed for sparse inputs, it surprisingly outperforms current SOTA methods in both full and sparse settings on the ZJU-MoCap and H36M datasets, demonstrating that our CHASE successfully maintains avatar's 3D consistency, hence improving rendering quality.

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.

Consistent 3D geometry estimation from streaming RGB input is crucial for real-world applications such as autonomous driving, embodied AI, and large-scale reconstruction. While modern monocular geometry foundation models achieve strong single-image accuracy, they exhibit severe temporal inconsistency on continuous input, notably dominated by scale--shift drifting. Through targeted empirical analysis, we trace this instability to its root cause: fluctuations in latent feature statistics, whose mean and variance directly determine the predicted depth's scale and shift. Building on this insight, we introduce Dynamic Feature Normalization (DyFN), a lightweight, causal recurrent module that dynamically and robustly modulates feature statistics to maintain stable geometry over time. We adapt powerful pretrained monocular geometry models for streaming by finetuning only DyFN, a mere 2\% additional parameters, while keeping the backbone frozen, thereby achieving temporal consistency without compromising single-image accuracy. Extensive experiments across four benchmarks show that DyFN effectively eliminates temporal artifacts such as disjointed layering and positional jitter, and achieves state-of-the-art temporal stability, improving over prior streaming methods by up to 14\% and even outperforming heavier non-causal video baselines. Project Page: https://shawlyu.github.io/DyFN

This is the second paper devoted to the numerical version of Signature-inverse Theorem in terms of the underlying joint invariants. Signature Theorem and its Inverse guarantee any application of differential invariant signature curves to the invariant recognition of visual objects. We first show the invalidity of Curvature-inverse and Signature inverse theorems, meaning non-congruent meshes may have the same joint invariant numerical curvature or signature. Then by classifying three and five point ordinary meshes respectively in the Euclidean and affine cases, we look for conditions in terms of the associated joint invariant signatures which make these theorems correct. Additionally, we bring forward The Host Theorem to provide a simpler version of Signature-inverse Theorem for closed ordinary meshes.

LLMs have demonstrated strong mathematical reasoning abilities by leveraging reinforcement learning with long chain-of-thought, yet they continue to struggle with theorem proving due to the lack of clear supervision signals when solely using natural language. Dedicated domain-specific languages like Lean provide clear supervision via formal verification of proofs, enabling effective training through reinforcement learning. In this work, we propose Seed-Prover, a lemma-style whole-proof reasoning model. Seed-Prover can iteratively refine its proof based on Lean feedback, proved lemmas, and self-summarization. To solve IMO-level contest problems, we design three test-time inference strategies that enable both deep and broad reasoning. Seed-Prover proves 78.1% of formalized past IMO problems, saturates MiniF2F, and achieves over 50\% on PutnamBench, outperforming the previous state-of-the-art by a large margin. To address the lack of geometry support in Lean, we introduce a geometry reasoning engine Seed-Geometry, which outperforms previous formal geometry engines. We use these two systems to participate in IMO 2025 and fully prove 5 out of 6 problems. This work represents a significant advancement in automated mathematical reasoning, demonstrating the effectiveness of formal verification with long chain-of-thought reasoning.

We propose a Geometry-aware Policy Imitation (GPI) approach that rethinks imitation learning by treating demonstrations as geometric curves rather than collections of state-action samples. From these curves, GPI derives distance fields that give rise to two complementary control primitives: a progression flow that advances along expert trajectories and an attraction flow that corrects deviations. Their combination defines a controllable, non-parametric vector field that directly guides robot behavior. This formulation decouples metric learning from policy synthesis, enabling modular adaptation across low-dimensional robot states and high-dimensional perceptual inputs. GPI naturally supports multimodality by preserving distinct demonstrations as separate models and allows efficient composition of new demonstrations through simple additions to the distance field. We evaluate GPI in simulation and on real robots across diverse tasks. Experiments show that GPI achieves higher success rates than diffusion-based policies while running 20 times faster, requiring less memory, and remaining robust to perturbations. These results establish GPI as an efficient, interpretable, and scalable alternative to generative approaches for robotic imitation learning. Project website: https://yimingli1998.github.io/projects/GPI/

Geometry is a fundamental branch of mathematics and plays a crucial role in evaluating the reasoning capabilities of multimodal large language models (MLLMs). However, existing multimodal mathematics benchmarks mainly focus on plane geometry and largely ignore solid geometry, which requires spatial reasoning and is more challenging than plane geometry. To address this critical gap, we introduce SolidGeo, the first large-scale benchmark specifically designed to evaluate the performance of MLLMs on mathematical reasoning tasks in solid geometry. SolidGeo consists of 3,113 real-world K-12 and competition-level problems, each paired with visual context and annotated with difficulty levels and fine-grained solid geometry categories. Our benchmark covers a wide range of 3D reasoning subjects such as projection, unfolding, spatial measurement, and spatial vector, offering a rigorous testbed for assessing solid geometry. Through extensive experiments, we observe that MLLMs encounter substantial challenges in solid geometry math tasks, with a considerable performance gap relative to human capabilities on SolidGeo. Moreover, we analyze the performance, inference efficiency and error patterns of various models, offering insights into the solid geometric mathematical reasoning capabilities of MLLMs. We hope SolidGeo serves as a catalyst for advancing MLLMs toward deeper geometric reasoning and spatial intelligence.

Generating long-range, geometrically consistent video presents a fundamental dilemma: while consistency demands strict adherence to 3D geometry in pixel space, state-of-the-art generative models operate most effectively in a camera-conditioned latent space. This disconnect causes current methods to struggle with occluded areas and complex camera trajectories. To bridge this gap, we propose WorldWarp, a framework that couples a 3D structural anchor with a 2D generative refiner. To establish geometric grounding, WorldWarp maintains an online 3D geometric cache built via Gaussian Splatting (3DGS). By explicitly warping historical content into novel views, this cache acts as a structural scaffold, ensuring each new frame respects prior geometry. However, static warping inevitably leaves holes and artifacts due to occlusions. We address this using a Spatio-Temporal Diffusion (ST-Diff) model designed for a "fill-and-revise" objective. Our key innovation is a spatio-temporal varying noise schedule: blank regions receive full noise to trigger generation, while warped regions receive partial noise to enable refinement. By dynamically updating the 3D cache at every step, WorldWarp maintains consistency across video chunks. Consequently, it achieves state-of-the-art fidelity by ensuring that 3D logic guides structure while diffusion logic perfects texture. Project page: https://hyokong.github.io/worldwarp-page/{https://hyokong.github.io/worldwarp-page/}.

Reasoning-trained language models often spend more tokens on harder problems, but longer chains of thought do not show whether a model is merely computing for more steps or following a different internal trajectory. We study this distinction through hidden-state trajectories during chain-of-thought generation across competitive programming, mathematics, and Boolean satisfiability. Raw trajectory geometry is strongly shaped by generation length: longer generations mechanically alter path statistics, so difficulty-dependent comparisons are misleading without adjustment. After residualizing trajectory statistics on length, difficulty remains systematically coupled to corrected trajectory geometry across all domains studied. The clearest reasoning-specific separation appears in the code domain, where harder problems show more direct corrected trajectories and less heterogeneous local curvature in reasoning-trained models than in matched instruction-tuned baselines. Corrected difficulty-geometry coupling is weaker, but still present, in mathematics and Boolean satisfiability. Prompt-stage linear probes do not mirror the code-domain separation, and behavioral annotations show that stronger corrected coupling co-occurs with strategy shifts and uncertainty monitoring. Together, these findings establish length correction as a prerequisite for generation-time trajectory analysis and show that reasoning training can be associated with distinct corrected trajectory geometry, with the strength of the effect depending on the domain.

Pretrained language models have shown superior performance on many natural language processing tasks, yet they still struggle at multi-step formal reasoning tasks like grade school math problems. One key challenge of finetuning them to solve such math reasoning problems is that many existing datasets only contain one reference solution for each problem, despite the fact that there are often alternative solutions resembling different reasoning paths to the final answer. This way, the finetuned models are biased towards the limited reference solutions, which limits their generalization to unseen examples. To mitigate this issue, we propose to let the model perform sampling during training and learn from both self-sampled fully-correct solutions, which yield the correct answer upon execution, and partially-correct solutions, whose intermediate state matches an intermediate state of a known correct solution. We show that our use of self-sampled correct and partially-correct solutions can benefit learning and help guide the sampling process, leading to more efficient exploration of the solution space. Additionally, we explore various training objectives to support learning from multiple solutions per example and find they greatly affect the performance. Experiments on two math reasoning datasets show the effectiveness of our method compared to learning from a single reference solution with MLE, where we improve PASS@100 from 35.5% to 44.5% for GSM8K, and 27.6% to 36.2% PASS@80 for MathQA. Such improvements are also consistent across different model sizes. Our code is available at https://github.com/microsoft/TraceCodegen.

We propose GeoUni, the first unified geometry expert model capable of generating problem solutions and diagrams within a single framework in a way that enables the creation of unique and individualized geometry problems. Traditionally, solving geometry problems and generating diagrams have been treated as separate tasks in machine learning, with no models successfully integrating both to support problem creation. However, we believe that mastery in geometry requires frictionless integration of all of these skills, from solving problems to visualizing geometric relationships, and finally, crafting tailored problems. Our extensive experiments demonstrate that GeoUni, with only 1.5B parameters, achieves performance comparable to larger models such as DeepSeek-R1 with 671B parameters in geometric reasoning tasks. GeoUni also excels in generating precise geometric diagrams, surpassing both text-to-image models and unified models, including the GPT-4o image generation. Most importantly, GeoUni is the only model capable of successfully generating textual problems with matching diagrams based on specific knowledge points, thus offering a wider range of capabilities that extend beyond current models.

Compact models often lose the structure of their embedding space. The issue shows up when the capacity is tight or the data spans several languages. Such collapse makes it difficult for downstream tasks to build on the resulting representation. Existing compression methods focus on aligning model outputs at a superficial level but fail to preserve the underlying manifold structure. This mismatch often leads to semantic drift in the compact model, causing both task behavior and linguistic properties to deviate from the reference model. To address those issues, we provide a new framework called Embedding Consistency Regulation (ECR). This framework first derives a set of semantic anchors from teacher embeddings (computed once offline). Then, the compact model learns to maintain consistent geometry around these anchors, without relying on matching logits or internal features. ECR adds only a small projection step at inference, without altering the decoding architecture or its runtime behavior. In experiments on a 100K multilingual corpus, ECR consistently stabilizes training and preserves semantic structure across tasks and languages. It also produces a more compact and task-aligned representation space, enabling low-capacity models to learn cleaner manifolds than conventional baselines. ECR works without teacher outputs and is compatible with, but independent of, distillation. Taken together, our results show that ECR helps compact models better follow task requirements and makes them easier to deploy under strict efficiency or privacy limits.

Multi-domain graph pre-training integrates knowledge from diverse domains to enhance performance in the target domains, which is crucial for building graph foundation models. Despite initial success, existing solutions often fall short of answering a fundamental question: how is knowledge integrated or transferred across domains? This theoretical limitation motivates us to rethink the consistency and transferability between model pre-training and domain adaptation. In this paper, we propose a fresh Riemannian geometry perspective, whose core idea is to merge any graph dataset into a unified, smooth Riemannian manifold, enabling a systematic understanding of knowledge integration and transfer. To achieve this, our key contribution is the theoretical establishment of neural manifold gluing, which first characterizes local geometry using an adaptive orthogonal frame and then "glues" the local pieces together into a coherent whole. Building on this theory, we present the GraphGlue framework, which supports batched pre-training with EMA prototyping and provides a transferability measure based on geometric consistence. Extensive experiments demonstrate its superior performance across diverse graph domains. Moreover, we empirically validated GraphGlue's geometric scaling law, showing that larger quantities of datasets improve model transferability by producing a smoother manifold. Codes are available at https://github.com/RiemannGraph/GraphGlue.

Using parts of existing models to rebuild new models, commonly termed as example-based modeling, is a classical methodology in the realm of computer graphics. Previous works mostly focus on shape composition, making them very hard to use for realistic composition of 3D objects captured from real-world scenes. This leads to combining multiple NeRFs into a single 3D scene to achieve seamless appearance blending. However, the current SeamlessNeRF method struggles to achieve interactive editing and harmonious stitching for real-world scenes due to its gradient-based strategy and grid-based representation. To this end, we present an example-based modeling method that combines multiple Gaussian fields in a point-based representation using sample-guided synthesis. Specifically, as for composition, we create a GUI to segment and transform multiple fields in real time, easily obtaining a semantically meaningful composition of models represented by 3D Gaussian Splatting (3DGS). For texture blending, due to the discrete and irregular nature of 3DGS, straightforwardly applying gradient propagation as SeamlssNeRF is not supported. Thus, a novel sampling-based cloning method is proposed to harmonize the blending while preserving the original rich texture and content. Our workflow consists of three steps: 1) real-time segmentation and transformation of a Gaussian model using a well-tailored GUI, 2) KNN analysis to identify boundary points in the intersecting area between the source and target models, and 3) two-phase optimization of the target model using sampling-based cloning and gradient constraints. Extensive experimental results validate that our approach significantly outperforms previous works in terms of realistic synthesis, demonstrating its practicality. More demos are available at https://ingra14m.github.io/gs_stitching_website.

We present GeoManip, a framework to enable generalist robots to leverage essential conditions derived from object and part relationships, as geometric constraints, for robot manipulation. For example, cutting the carrot requires adhering to a geometric constraint: the blade of the knife should be perpendicular to the carrot's direction. By interpreting these constraints through symbolic language representations and translating them into low-level actions, GeoManip bridges the gap between natural language and robotic execution, enabling greater generalizability across diverse even unseen tasks, objects, and scenarios. Unlike vision-language-action models that require extensive training, operates training-free by utilizing large foundational models: a constraint generation module that predicts stage-specific geometric constraints and a geometry parser that identifies object parts involved in these constraints. A solver then optimizes trajectories to satisfy inferred constraints from task descriptions and the scene. Furthermore, GeoManip learns in-context and provides five appealing human-robot interaction features: on-the-fly policy adaptation, learning from human demonstrations, learning from failure cases, long-horizon action planning, and efficient data collection for imitation learning. Extensive evaluations on both simulations and real-world scenarios demonstrate GeoManip's state-of-the-art performance, with superior out-of-distribution generalization while avoiding costly model training.

Developing a unified pipeline that enables users to remove, re-texture, or replace objects in a versatile manner is crucial for text-guided 3D inpainting. However, there are still challenges in performing multiple 3D inpainting tasks within a unified framework: 1) Single reference inpainting methods lack robustness when dealing with views that are far from the reference view. 2) Appearance inconsistency arises when independently inpainting multi-view images with 2D diffusion priors; 3) Geometry inconsistency limits performance when there are significant geometric changes in the inpainting regions. To tackle these challenges, we introduce DiGA3D, a novel and versatile 3D inpainting pipeline that leverages diffusion models to propagate consistent appearance and geometry in a coarse-to-fine manner. First, DiGA3D develops a robust strategy for selecting multiple reference views to reduce errors during propagation. Next, DiGA3D designs an Attention Feature Propagation (AFP) mechanism that propagates attention features from the selected reference views to other views via diffusion models to maintain appearance consistency. Furthermore, DiGA3D introduces a Texture-Geometry Score Distillation Sampling (TG-SDS) loss to further improve the geometric consistency of inpainted 3D scenes. Extensive experiments on multiple 3D inpainting tasks demonstrate the effectiveness of our method. The project page is available at https://rorisis.github.io/DiGA3D/.

Interactive world models continually generate video by responding to a user's actions, enabling open-ended generation capabilities. However, existing models typically lack a 3D representation of the environment, meaning 3D consistency must be implicitly learned from data, and spatial memory is restricted to limited temporal context windows. This results in an unrealistic user experience and presents significant obstacles to down-stream tasks such as training agents. To address this, we present PERSIST, a new paradigm of world model which simulates the evolution of a latent 3D scene: environment, camera, and renderer. This allows us to synthesize new frames with persistent spatial memory and consistent geometry. Both quantitative metrics and a qualitative user study show substantial improvements in spatial memory, 3D consistency, and long-horizon stability over existing methods, enabling coherent, evolving 3D worlds. We further demonstrate novel capabilities, including synthesising diverse 3D environments from a single image, as well as enabling fine-grained, geometry-aware control over generated experiences by supporting environment editing and specification directly in 3D space. Project page: https://francelico.github.io/persist.github.io

We present a method for 3D reconstruction only using calibrated multi-view surface azimuth maps. Our method, multi-view azimuth stereo, is effective for textureless or specular surfaces, which are difficult for conventional multi-view stereo methods. We introduce the concept of tangent space consistency: Multi-view azimuth observations of a surface point should be lifted to the same tangent space. Leveraging this consistency, we recover the shape by optimizing a neural implicit surface representation. Our method harnesses the robust azimuth estimation capabilities of photometric stereo methods or polarization imaging while bypassing potentially complex zenith angle estimation. Experiments using azimuth maps from various sources validate the accurate shape recovery with our method, even without zenith angles.

Estimating the pose of an object from a monocular image is an inverse problem fundamental in computer vision. The ill-posed nature of this problem requires incorporating deformation priors to solve it. In practice, many materials do not perceptibly shrink or extend when manipulated, constituting a powerful and well-known prior. Mathematically, this translates to the preservation of the Riemannian metric. Neural networks offer the perfect playground to solve the surface reconstruction problem as they can approximate surfaces with arbitrary precision and allow the computation of differential geometry quantities. This paper presents an approach to inferring continuous deformable surfaces from a sequence of images, which is benchmarked against several techniques and obtains state-of-the-art performance without the need for offline training.

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

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.

Recent advances in generative models highlight the power of geometry-aware modeling in manifold-constrained settings. Yet, for natural images, the field remains confined to Euclidean assumptions, failing to exploit the potential of intrinsic geometric structures within the data. In this work, we investigate the geometry of natural images and observe that semantic information is predominantly encoded in directional components, while norm components can be approximated by the global average. This property holds across both RGB and latent spaces, suggesting that natural images can be effectively modeled on a hypersphere. Building on this finding, we introduce Spherical Optimal Transport Flow Matching (SOT-CFM), which utilizes angular distance, and Spherical Flow Matching (SFM), which constrains dynamics directly on the manifold. Our experiments demonstrate that these geometry-aware methods achieve superior performance against Euclidean baselines. Ultimately, this work provides a novel perspective that bridges the gap between Riemannian manifold-based modeling and natural image generation.

Dense matching methods like DUSt3R regress pairwise pointmaps for 3D reconstruction. However, the reliance on pairwise prediction and the limited generalization capability inherently restrict the global geometric consistency. In this work, we introduce Test3R, a surprisingly simple test-time learning technique that significantly boosts geometric accuracy. Using image triplets (I_1,I_2,I_3), Test3R generates reconstructions from pairs (I_1,I_2) and (I_1,I_3). The core idea is to optimize the network at test time via a self-supervised objective: maximizing the geometric consistency between these two reconstructions relative to the common image I_1. This ensures the model produces cross-pair consistent outputs, regardless of the inputs. Extensive experiments demonstrate that our technique significantly outperforms previous state-of-the-art methods on the 3D reconstruction and multi-view depth estimation tasks. Moreover, it is universally applicable and nearly cost-free, making it easily applied to other models and implemented with minimal test-time training overhead and parameter footprint. Code is available at https://github.com/nopQAQ/Test3R.

Our world is full of identical objects (\emphe.g., cans of coke, cars of same model). These duplicates, when seen together, provide additional and strong cues for us to effectively reason about 3D. Inspired by this observation, we introduce Structure from Duplicates (SfD), a novel inverse graphics framework that reconstructs geometry, material, and illumination from a single image containing multiple identical objects. SfD begins by identifying multiple instances of an object within an image, and then jointly estimates the 6DoF pose for all instances.An inverse graphics pipeline is subsequently employed to jointly reason about the shape, material of the object, and the environment light, while adhering to the shared geometry and material constraint across instances. Our primary contributions involve utilizing object duplicates as a robust prior for single-image inverse graphics and proposing an in-plane rotation-robust Structure from Motion (SfM) formulation for joint 6-DoF object pose estimation. By leveraging multi-view cues from a single image, SfD generates more realistic and detailed 3D reconstructions, significantly outperforming existing single image reconstruction models and multi-view reconstruction approaches with a similar or greater number of observations.