Geometry-Aware

Optimal transport gives structured data a geometry, but exact evaluation is costly in large pairwise analyses that exploit relationships among distances. Learned surrogates are faster, but need not preserve this metric structure. We develop a Deep Sets surrogate for OT between variable-size weighted point clouds that enforces non-negativity, exchange symmetry, and zero self-distance, leaving the triangle inequality unconstrained. Applied to the Energy Mover's Distance between collider events in a particle physics application, the Metric-Aware Particle Flow Network achieves percent-level mean absolute percentage error while significantly improving inference throughput over other exact and approximate methods surveyed. The architectural constraints are found to improve properties that are not explicitly enforced: across $10^6$ held-out event triplets, triangle-inequality violations fall from 199 for a matched unconstrained network to 2, and the maximum from 149.5 to 5.8 GeV. These results demonstrate that targeted inductive biases can yield fast neural surrogates with substantially improved geometric fidelity.

Activation steering provides a lightweight way to control large language models (LLMs) by modifying their hidden activations at inference time. Among these approaches, norm-preserving steering aims to change model behavior without altering the activation norm, reducing the risk of representation collapse and degradation. However, existing norm-preserving methods are limited by predefined steering trajectories and by their reliance on one-step updates, which may fail to capture the complex structure of activation distributions. We propose GeoSteer, an optimization-based method for norm-preserving activation steering. GeoSteer formulates steering as a Riemannian optimization problem and updates activations through a sequence of small geodesic steps on the representation manifold. To avoid fixed steering directions, GeoSteer learns a nonlinear activation-space objective that distinguishes desired from undesired activations, and uses this function to adaptively guide each steering step. This multistep formulation yields smoother, more stable, and more consistent steering behavior while preserving the activation norm. Across TruthfulQA, RealToxicityPrompts, and UltraFeedback benchmarks, GeoSteer consistently improves over state-of-the-art activation steering baselines. These results suggest that norm-preserving steering can be made more effective by replacing predefined one-step edits with adaptive, geometry-aware optimization.

Several geometry-aware approaches to low-rank adaptation have emerged for parameter-efficient fine-tuning of large pre-trained models. These methods aim to take full advantage of the geometric structure of low-rank manifolds for improving the efficiency in subspace utilization and reducing redundancy by enforcing orthogonality constraints during optimization. The strong empirical results of these techniques have motivated further study into whether predictions from such geometry-based adaptation methods could be overconfident. In this paper, we build on the singular value decomposition factorization of adapters to develop a framework based on Stein variational gradient descent (SVGD). In this formulation, the low-rank matrices are transported along the Stiefel manifold to match the targeted distributions while retaining their crucial geometric structure. Since this geometry-aware SVGD approach provides multiple solutions during inference, it supports uncertainty quantification and produces better-calibrated adapters on the Stiefel manifold. Extensive experiments show that our method delivers strong model calibration and attains higher prediction accuracy than SVGD and related uncertainty estimation methods that are formulated in Euclidean space.

Estimating the 6D pose of textureless objects without prior CAD models remains a critical challenge due to the lack of appearance features. While recent generalizable approaches alleviate the dependence on object-specific models, their performance on low-texture objects is often limited by insufficient geometric constraints in the underlying representations. In this work, we propose PG-Pose, a geometry-aware framework combining Planar-based Gaussian Splatting (PGS) reconstruction and Geometry-driven pose optimization. In the offline representation extraction stage, three distinct representations of the object are extracted from multi-view reference RGB images with known poses. PG-Pose reconstructs a 3D Gaussian representation and renders high-fidelity depth maps to generate 3D point clouds through back projection. In the online pose inference stage, the initial pose of the input image is estimated by 2D-3D correspondence matching between the input image and the reconstructed 3D point clouds, followed by a PGS-Refiner for iterative pose optimization. Evaluations on the OnePose-LowTexture datasets, PG-Pose achieves an average accuracy of 94.2% ADD(S)@0.1d, with a 2.1% improvement average accuracy compared with the state-of-the-art (SOTA) GS-based approach. To further demonstrate the effectiveness of PG-Pose for industrial robots in grasping tasks, we deploy it on a dual-arm industrial robot and successfully realize the grasping task on an unseen object.

Multi-agent combinatorial optimization problems are notoriously challenging due to their NP-hard nature. Recent parallel autoregressive neural solvers improve inference efficiency by allowing agents to make decisions simultaneously, but their performance often degrades on large-scale instances. This is largely attributable to weak modeling of local geometric structures and the fact that conflicting task selections are handled only after action generation. To address these limitations, we propose GeoPAR, a geometry-guided parallel autoregressive reinforcement learning framework for scalable multi-agent combinatorial optimization. GeoPAR integrates three key components: (1) a projection-window sparse geometry mechanism that builds lightweight local candidate neighborhoods through multi-directional projections, (2) sparse edge-biased attention that injects these geometric relations into node representations, and (3) cache-guided conflict-aware assignment that reuses the geometric cache during decoding to suppress duplicate selections of exclusive tasks. Experiments on heterogeneous vehicle routing and open multi-depot pickup-and-delivery problems show that GeoPAR improves large-scale zero-shot generalization while substantially reducing rollout steps and maintaining efficient inference.

Solving multiphysics partial differential equations (PDEs) remains a major challenge in scientific computing, especially for highly complex $μ$m-scale tortuous geometries critical to energy and chemical engineering. We address this challenge by proposing a Geometry-aware Latent Autoregressive generative Model for PDEs (GeoLAMP) for solving physics within highly irregular and tortuous structures. GeoLAMP introduces a dual-encoder architecture on graph representations to jointly capture global topology and fine-scale geometric features, enabling an effective transition from real-space fields to compact latent representations. In the latent space, we propose a causal self-attention transformer with flow matching to model temporal dynamics, allowing stable and scalable block-wise autoregressive prediction. A flexible decoder reconstructs high-resolution physical fields on arbitrary points. We establish three multiphysics benchmark datasets in complex geometries, covering reactive flow, heat convection, and elasticity. GeoLAMP consistently achieves the most stable autoregression performance on these datasets, maintaining low errors throughout the entire rollout horizon. Our results provide a systematic study of geometry-aware learning for PDEs in $μ$m-scale complex geometries and offer new insights into block-wise time marching of latent autoregressive PDE modeling via a flow matching framework.

Autoregressive large models have recently advanced Text-to-SVG generation from simple icons to complex, long-context graphics, yet standard autoregressive decoding often fails to maintain global consistency across geometry, layout, occlusion, and composition. We introduce CANVAS (Consistency-Aware Navigation via Visual Adaptive Sampling), a training-free, render-aware inference framework that combines power-sharpened trajectory likelihood with visual feedback from rendered futures and derives a stroke-wise navigation rule. It effectively estimates each candidate stroke's future value under a limited generation and rendering budget and adaptively allocates samples according to candidate uncertainty, decision influence, and rollout cost. Experiments across multiple autoregressive SVG backbones and complementary benchmarks demonstrate improvements in global consistency, which includes sound geometric relationships, spatial layouts, occlusion ordering, and overall composition, without additional training, demonstrating the effectiveness and generalization ability of our framework.

Geometry-aware video object scaling aims to anisotropically resize the object along object-centric axes while preserving geometric plausibility, temporal coherence, and background consistency. Existing text-guided methods mainly operate in the 2D image plane, while depth-guided approaches provide coarse control and mesh-based methods require costly 3D reconstruction. We present a progressive two-stage training framework that decouples geometry-aware foreground transformation from background preservation and realistic video composition, without mesh-pixel alignment and explicit 3D reconstruction at inference. In both stages, geometrically perturbed pseudo-sources are constructed from real videos, while the original complete videos are retained as reconstruction targets. The first stage uses planar transformations to learn robust foreground-background composition, whereas the second introduces object-centric 3D deformation guidance for geometry-aware scaling. This pseudo-source reconstruction formulation enables real-video synthesis without paired real-world scaling targets. We construct complementary paired-geometry and real-background benchmarks and further evaluate on in-the-wild videos. Extensive experiments demonstrate superior geometric consistency, foreground fidelity, and background preservation, together with faster and more practical inference than methods requiring explicit 3D reconstruction.

Foundation models are transforming business workflows and boosting productivity, yet they remain largely absent from engineering domains such as power system analysis, where strict physical consistency must be enforced. We present GENCO (GEometric Neural Corrective Optimizer), a unified neural solver for steady-state transmission grid analysis that handles power flow (PF), optimal power flow (OPF), and state estimation (SE) within a single architecture and shared network representation. To support advances in neural power system solvers, we introduce the open-source GridFM Development Framework, which standardizes synthetic data generation and training in a low-code environment. We also release large-scale datasets with millions of PF and OPF scenarios across diverse grid topologies to support reproducible benchmarking. We evaluate GENCO on the PFDelta and OPFData benchmarks against state-of-the-art neural solvers and classical solvers, including Newton-Raphson and IPOPT, as well as on real-world Hydro-Québec SCADA data. For large-scale PF, GENCO recovers the full AC operating state, including voltage magnitudes and reactive power that DC-PF cannot provide, while matching DC-PF-level active power-balance residuals. It achieves up to 30x speedups over Newton-Raphson at only 2x the runtime of DC-PF. For OPF, it achieves up to 85x speedups over IPOPT while improving feasibility, optimality, and runtime over DC-OPF. For SE, GENCO is more robust than classical weighted least squares to noisy measurements and network parameter errors, and always returns a high-quality estimate even when weighted least squares fails to converge. Together, the unified architecture and development framework provide a new approach to large-scale steady-state grid analysis, lowering the barrier to entry for power system engineers and marking a step toward Grid Foundation Models.

Long-horizon future-frame prediction is important for autonomous driving, traffic surveillance, and intelligent transportation systems, yet remains challenging due to temporal ghosting, geometry drift, and inconsistent object motion. Recent latent video diffusion models have achieved impressive visual quality, but directly applying them to structured traffic scenes often leads to unstable geometry and degraded temporal coherence over extended horizons. We present a training-free inference framework that stabilizes reliable static structure in pretrained video predictions through multi-frame temporal context and view-conditioned routing. For front-camera videos, our method refines generated futures with a multi-frame depth-layered renderer that projects static geometry from observed history frames while preserving dynamic regions from the generative base model. For heterogeneous traffic views, a frozen vision-language model infers a coarse camera group from the observed clip and selects a specialized motion-based predictor. The framework requires neither retraining nor fine-tuning of the underlying video model and can be applied directly to pretrained generators. We validate the proposed framework on the AI City Challenge Track 5 benchmark, where our final system achieves competitive performance among the top-ranked teams. These results demonstrate that geometry-aware inference-time refinement and view-conditioned hybrid inference can improve static-geometry stability and low-level structural fidelity without changing the original model architecture.

Learning-based wireless localizers often fail to utilize geometric information about the propagation environment, limiting their ability to exploit non-line-of-sight (NLoS) propagation and generalize across scenes. To bridge this gap, we propose GLocFM, a Geometry-aware Localization Foundation Model, which jointly exploits WiFi measurements and scene geometry represented as a 3D point cloud. We formulate localization as a maximum-likelihood (ML) estimation problem, where the goal is to find a transmitter position that maximizes the likelihood of the wireless observations conditioned on the scene geometry. The likelihood of a candidate transmitter position is calculated by a learned scoring function that matches the observed delay--angle-of-arrival (AoA) spectrum against the spectrum predicted for that candidate. A hierarchical scene encoder extracts propagation-relevant features to produce geometric priors for LoS and one-bounce reflection paths. For scenarios with imperfect synchronization, we further introduce a time-of-flight (ToF)-robust GLocFM model to handle unknown ToF offsets. GLocFM is trained on a multi-modal synthetic indoor localization dataset comprising 221 diverse scenes whose associated wireless signals are generated using Sionna RT. On both synthetic and the NeRF$^{2}$ dataset based on real measurements, GLocFM reduces mean 3D localization error relative to one of the state-of-the-art localization baselines by 49.5% and 48.8%, respectively. Ablations across different number of receiver, bandwidths, and array sizes further demonstrate the effectiveness and robustness of the proposed framework.

2D affordance grounding aims to locate the region of an object that a human can interact with. Existing research focuses on recognizing affordance types seen during training and does not study models' ability to generalize to novel affordances, which is crucial for real-world applications. We propose the task of zero-shot 2D grounding with novel affordance types (NAT) and introduce the NAT benchmarks. We then propose AffordAnything, a training-free method that leverages segmentation cues, motivated by the strong correlation between affordance regions and object subparts. To further improve performance, we develop AffordAnything+, a trainable variant that learns to combine these cues. On the proposed AGD20K-NAT benchmark, our best model AffordAnything+ achieves a substantial improvement of 12.3% (absolute) in IoU@0.4 over the SOTA affordance grounding method, OOAL.

Fixed-layout indoor furniture styling requires selecting assets that form a coherent room without changing the prescribed furniture categories, positions, orientations, or scales. Existing approaches typically retrieve each asset independently or rely on static local relations, making them prone to shape, material, and color conflicts after scene composition. We introduce StyleForge, a scene-level structured selection framework built on a dynamic hypergraph style field. A frozen multimodal large language model extracts structured style priors from an open-ended style request and the fixed layout, while StyleForge maintains a learnable candidate distribution for each furniture slot. Conditioned on the target style, the dynamic hypergraph style field adaptively activates and weights layout-induced hyperedges to capture higher-order dependencies among furniture. Counterfactual style preference learning then treats each candidate as a local substitution in the current style field and evaluates its contextual compatibility using Mahalanobis energies. Training alternates between optimizing the style field and the candidate logits. At inference, the model remains frozen and test-time training updates only room-specific candidate logits, progressively correcting cross-slot style conflicts as the global scene context evolves. Experiments on 3D-FRONT demonstrate state-of-the-art furniture retrieval and scene-level style coherence, producing more coherent fixed-layout furniture arrangements than object- and scene-level retrieval baselines.

Existing generative models for earth observation (EO) predominantly rely on fine-tuning natural image priors, which limits their scalability and introduces perspective biases that conflict with geospatial constraints. To address this, we introduce GeoCore-9B, a 9-billion-parameter generative foundation model, which is the first of its scale to be trained from scratch exclusively on EO data. Unlike previous EO foundation models, GeoCore-9B is built upon a Flow Matching-based Diffusion Transformer (DiT) and natively conditions generation on text descriptions and continuous geospatial metadata, including ground sample distances, latitudes, and longitudes. To overcome the convergence and spatial disorientation challenges of training at this scale, we propose a Geospatial Semantic Alignment loss. This objective distills structural Earth surface priors (e.g., terrain and urban areas) from a frozen specialist teacher network, constraining the diffusion latent trajectory during training without adding inference overhead. Pre-trained on the global-scale Git-10M dataset, GeoCore-9B demonstrates strong downstream versatility. Beyond standard proxy generative tasks, we show that GeoCore-9B can be effectively adapted for practical EO applications, including highly challenging tasks such as cloud removal and SAR-to-optical cross-modal translation. Extensive evaluations confirm that GeoCore-9B establishes new state-of-the-art performance in both visual fidelity and geographic structural accuracy.

Neural operators have shown strong potential for learning solution operators of partial differential equations (PDEs). However, long-horizon autoregressive prediction remains challenging: local errors accumulate as spectral inconsistency, phase misalignment, or mean drift. Existing methods mainly improve state representations and operator backbones, while leaving the repeatedly applied latent transition increment weakly structured, allowing spectral errors and unstable channel couplings to accumulate during rollout. To address these issues, we propose a geometry-aware incremental neural operator (GeoIncNO) for stable long-horizon PDE prediction. GeoIncNO predicts latent increments for residual advancement and uses lightweight low-rank projectors to regulate channel coupling within active frequency bands derived from the increment spectral energy distribution. To reduce physical-space reconstruction errors, GeoIncNO further introduces a mean--fluctuation decoupled reconstruction mechanism, where stable mean structures and dynamic fluctuations are fused separately, and phase correction is applied only to the zero-mean fluctuation component. Extensive experiments on six PDE benchmarks, covering 1D, 2D, and 3D dynamical systems, show that GeoIncNO achieves consistently strong prediction accuracy, improved rollout stability, and better spectral fidelity compared with competitive neural-operator baselines.

Existing drone-view geo-localization (DVGL) methods are mainly developed under a static training paradigm, where models are optimized for fixed environments with all training data available in advance. However, this paradigm is difficult to extend to real-world deployment, where drones may encounter diverse environments and require multiple environment-specific models, resulting in additional storage and model-selection costs. Directly adapting a single model to new environments also risks distorting previously learned cross-view embedding geometry and causing forgetting. To address these challenges, we formalize the continual drone-view geo-localization (C-DVGL) setting and propose GeoMFD, a geometry-aware continual adaptation method for DVGL. GeoMFD combines a cold-start bootstrapping strategy (CBS), a geometry-aware adapter (Geo-Adapter), and margin-field distillation (MFD) to balance adaptation and cross-view geometry preservation. CBS initializes a stable embedding space, Geo-Adapter enables environment adaptation through controlled residual corrections, and MFD preserves similarity margins between positive pairs and hard negatives to alleviate cross-view geometry forgetting. Extensive experiments demonstrate that GeoMFD effectively mitigates forgetting and achieves competitive performance with environment-specific DVGL methods using a single continuously updated model.

Large pretrained vision models have substantially improved appearance-based 3D shape retrieval, but they still confuse shapes that look similar while differing in geometry. Although geometry-aware features can reduce these errors, naive fusion of geometry and appearance may hurt retrieval when the two modalities are already well aligned. We propose GATE-3D, a lightweight query-adaptive reranking method that incorporates geometry without retraining the retrieval backbone. For each query, GATE-3D predicts how much a geometry-aware score should adjust the appearance-based ranking using features that capture disagreement between the two modalities. This selective design lets geometry contribute where it helps and stay silent where it would hurt. Experiments on three open-set 3D retrieval benchmarks show that GATE-3D improves over appearance-only retrieval and is more robust than always-on fusion. On the primary benchmark, it improves mAP@10 by 2.00 points over appearance-only retrieval (p=0.041); it also improves leave-one-category-out generalization and reduces geometric false positives by 10.8%. GATE-3D achieves competitive zero-shot results against DAC-based baselines. We further find that simple linear routing is more effective than a small MLP in the low-data regime, suggesting that cross-modal disagreement features matter more than model capacity for adaptive routing.

Multi-channel speech enhancement (SE) systems exhibit superior performance over single-channel methods but are constrained to fixed microphone array configurations. This restricts their real-world deployment across devices with diverse array geometries. While recent array-agnostic SE methods address variable microphone numbers and permutations, they largely fail to exploit explicit array geometry priors when available, missing a crucial cue for optimal spatial filtering. A Geometry-Aware Dynamic Convolution (Geo-DConv) framework is proposed, which explicitly leverages microphone coordinates to transform standard fixed-array SE models into robust array-invariant systems. Experiments are conducted on the recent real-recorded RealMAN multi-channel speech dataset. Results demonstrate that the proposed architecture enables two widely used fixed-array models to adapt to array-invariant settings, with consistent performance improvements across diverse array topologies.

Existing 3D generative models predominantly rely on implicit volumetric representations, which enforce watertight topology and struggle to represent thin-shell and non-manifold geometries such as garments. Geometry image-based approaches offer a surface-centric alternative, but existing methods rely on discrete binary occupancy maps whose resolution-dependent boundary encoding causes staircase artifacts and information loss upon downsampling, while surface reconstruction remains a non-differentiable post-processing step disconnected from the learning pipeline. To address this, we propose Differentiable Geometry Image (DiffGI), an end-to-end 3D-to-2D mapping framework that seamlessly integrates surface representation and geometric optimization. DiffGI replaces binary maps with a continuous 2D Truncated Signed Distance Function (TSDF), which encodes boundary position at subpixel precision within a fixed grid resolution, eliminating resolution-dependent staircase artifacts even under aggressive downsampling. Building on this continuous field, we introduce a differentiable Marching Squares algorithm based on analytical linear interpolation, allowing gradients from 3D surface losses to propagate back to the 2D latent space. Leveraging this differentiable pipeline, we train a DiffGI-VAE augmented with a geometry-aware normal rendering loss to compress complex 3D surfaces into an ultra-compact 32X32 latent space, and instantiate a transformer-based latent diffusion model with a flow-matching objective on top of this space for conditional 3D generation. Extensive experiments on garment and object datasets demonstrate that our method achieves superior reconstruction fidelity and boundary precision compared to prior geometry-image and voxel-based approaches, while requiring significantly fewer computational resources.

How can an agent build a structured map of its world from nothing but an ongoing sequence of raw sensory input and its own movements, especially when natural variation means exact sensory patterns rarely repeat? The Clone-Structured Causal Graph algorithm (CSCG), a normative hippocampus model, shows how an interpretable map can be learned from aliased observations. However, CSCG requires a predefined discrete alphabet, and its expectation-maximization formulation is not easily combined with existing neural network modules, preventing the end-to-end processing of raw image sequences. We remove this barrier by reformulating CSCG as a single, fully differentiable module, gradCSCG, and coupling it to a learned vector-quantized variational autoencoder (VQ-VAE) perceptual front-end. A soft emission forward pass allows the map-learning objective to flow back into perception, while a set of loss-balancing mechanisms mitigates module collapse during joint training. We demonstrate, first, that gradient training reproduces CSCG's results on original symbolic grid worlds by recovering room topology from heavily aliased observations. Second, we show that map recovery remains robust on MNIST image sequences, where each visit to a location yields a newly sampled image of its assigned digit. Across four heavily aliased environments, the end-to-end pipeline successfully uncovers the underlying adjacency graph with high edge precision and recall, directly from visual input. This work provides a proof of principle that CSCG can serve as a composable building block in a deep learning architecture.

Reliable visual data association is fundamental to visual SLAM (V-SLAM), as it directly determines the quality of the camera pose estimation and map consistency. However, the handcrafted descriptors used by most mature real-time systems degrade under illumination and viewpoint changes, while learning-based front-ends that address this weakness typically require replacing the extraction-and-matching pipeline and introduce substantial computational overhead. Descriptor enhancement offers a compromise by refining existing descriptors within their original format, yet current methods rely on simplified attention mechanisms whose limited contextual modeling constrains the achievable matching quality. To resolve this trade-off between contextual expressiveness and efficiency, we propose Desc++, a lightweight enhancement module that jointly encodes descriptor representations and keypoint geometry and aggregates spatial context through a hybrid architecture that combines order-agnostic global attention with geometry-aware sequential modeling in linear time. The enhanced descriptors retain their original dimensionality and matching interface, enabling integration into deployed V-SLAM systems without modifying the pipeline. Experiments across descriptor matching, correspondence analysis, and system-level benchmarks with four different V-SLAM systems demonstrate that Desc++ improves matching accuracy over the state-of-the-art enhancement method, translates these gains into more accurate and stable trajectory estimation, and achieves a favorable balance between accuracy and efficiency for practical integration into existing real-time V-SLAM pipelines.

Accurate cervical cytology image classification is a key component of automated cervical cancer screening, where reliable recognition of normal, precancerous, and cancer-associated cellular patterns from Pap smear images can improve screening efficiency and diagnostic consistency. However, this task remains challenging because cervical cells exhibit complex morphology, subtle intra-class variations, and strong inter-class similarities. Existing convolution-based models capture local texture well but have limited ability to model long-range relationships, whereas attention-based models provide broader context but often lack explicit structural guidance. To address these limitations, we propose a geometry-aware classification framework for cervical cancer screening-oriented cytology image analysis, incorporating semantic abstraction and structural priors learned from pre-trained vision-language features. The method uses Gaussian expert modules to generate axis-wise priors from global semantic information, capturing structural regularities such as nuclear alignment and cellular spatial organization. These priors are embedded into an axial self-attention module to modulate similarity computation along horizontal and vertical directions, improving long-range dependency modeling and structure-sensitive feature interaction. Experiments on the Mendeley liquid-based cytology and SIPaKMeD datasets show that the proposed method achieves 99.48% accuracy on the former and 96.08% on the latter, with balanced gains in recall, precision, and overall classification performance. Visual analysis further shows that the learned priors highlight diagnostically relevant cellular regions, demonstrating the potential of the proposed framework as a screening-oriented decision-support tool for cervical cytology.

In this work, we present Canvas360, a two-stage framework for in-context panoramic generation that combines geometry-aware pretraining with downstream task-specific fine-tuning. To address the lack of large-scale, high-quality training data tailored to in-context panoramic tasks, we propose Canvas360Dataset, a collection of 1M high-quality paired panoramic samples for style transfer, inpainting, outpainting, and editing, enabling effective supervision across diverse in-context generation scenarios. On the modeling side, Canvas360 enhances text-to-panorama generation through parallel depth generation, velocity circular padding, and similarity loss regularization, enabling the model to learn geometry-aware representations, capture object distortion details, and improve geometric consistency and global coherence. Furthermore, empowered by strong panoramic priors, Canvas360 enables a unified in-context panoramic generation framework that supports diverse downstream tasks via token-level concatenation, surpassing prior methods in both task coverage and modeling flexibility. Extensive experiments show that Canvas360 improves panoramic image fidelity, achieving particularly strong performance on the panorama-specific FAED metric and competitive or leading results across the reported quantitative evaluations. More information can be found on our project page: https://zry000.github.io/Canvas360/

Navigational bronchoscopy is critical for pulmonary interventions, yet current platforms depend heavily on pre-operative CT or external sensors, limiting their use in critical care and resource-constrained settings. Vision-only navigation offers a scalable alternative, but conventional visual odometry (VO) struggles with texture-poor airway images, specularities, and the vanishing-point singularities of tubular anatomy, leading to frequent tracking failures and drift. We present a geometry-aware VO framework that explicitly leverages vanishing-point cues from airway lumens. Detected lumens are back-projected to 3D rays, whose weighted fusion yields a stable forward heading even when parallax cues are absent. This heading, together with looming-based velocity estimates, is fused with noisy VO outputs using a bespoke high-gain observer that enforces airway-following priors and rejects drift. We validate the method on ex-vivo mechanically ventilated human lungs with electromagnetic tracking ground truth. Compared to state-of-the-art pipelines (ORB-SLAM2, LoFTR-VO, DPVO), our approach reduces absolute trajectory error by more than 50% and achieves the lowest relative pose error across all test sequences.

Low-light degradation reduces image visibility and weakens structural cues that are important for visual representation and scene understanding. Existing low-light image enhancement methods mainly focus on appearance restoration, while insufficiently exploiting scene geometry to preserve structural consistency. To address this limitation, this paper proposes a Depth-guided Multi-scale Attention Network (DMSA-Net) for geometry-aware low-light image enhancement. DMSA-Net introduces depth-related structural priors into low-light representation learning through reflectance-geometry interaction. A Retinex-based decomposition module is first used to obtain illumination-invariant reflectance representations, from which depth cues are inferred to characterize scene structure under degraded illumination. A multi-scale depth-guided fusion strategy is then embedded into a hierarchical encoder-decoder architecture, where depth-aware attention adaptively integrates geometric and appearance features. Experiments on several benchmark datasets show that DMSA-Net achieves effective low-light restoration while improving structural preservation. Moreover, we construct LOL-D, a depth-augmented low-light dataset, to facilitate research on geometry-aware low-light vision.

Accurately estimating the unknown target label distribution is the critical first step for adapting to label shift. This task, widely known as quantification or class prevalence estimation, has recently seen significant advances through continuous KDE-based methods which model the density of multiclass classifier posteriors. Posterior vectors might be regarded as compositional data, since they lie on the probability simplex. However, existing KDE-based quantifiers typically rely on Euclidean Gaussian kernels, which ignore simplex geometry and incorrectly assign probability mass outside its boundaries. We introduce a geometry-aware KDE model for multiclass quantification based on log-ratio representations and Aitchison geometry, together with a shrinkage regularization that improves robustness near the simplex boundary. Combined with a maximum-likelihood interpretation of KDE-based quantification, we derive both point-estimation and Bayesian inference procedures for class prevalences. Experiments on 42 datasets across tabular, text, and image domains show that the proposed method is competitive with state-of-the-art quantifiers, often improving over standard KDE-based baselines, while also yielding strong results among Bayesian quantification methods.

Accurate work-zone geometry perception is critical for intelligent transportation systems, and ultra-wideband sensing offers a low-cost approach for infrastructure-aided reconstruction. However, outdoor UWB ranging is often degraded by non-line-of-sight propagation, burst noise, and long-tail errors, which can distort downstream spatial reconstruction. We present GAIA, a geometry-aware, infrastructure-anchored learning framework that couples temporal range modeling with latent anchor-layout estimation and deterministic distance projection. GAIA preserves range denoising as the supervised task while orienting the learned distances toward boundary-consistent reconstruction. We evaluate GAIA on a real-world outdoor UWB dataset with synchronized UWB, GNSS, and IMU measurements, and further test robustness using a real-data-calibrated stress-test simulator. GAIA achieves the lowest overall range MSE and highest polygon IoU among evaluated filtering-based and learning-based baselines, reducing MSE by 18.4% and improving polygon IoU by 15.5% over PoseMLP. These results show that geometry-aware range denoising provides an effective path toward spatially coherent work-zone reconstruction.

Recent Multi-modal Large Language Models (MLLMs) have demonstrated remarkable performance on 2D question answering tasks. However, extending these models to the 3D question answering remains challenging, as they typically require multiple views of the scene, which incurs substantial computational cost at inference. To mitigate this issue, existing solutions rely on strategic frame selection or token-merging algorithms that require preprocessing in advance all frames of the scene, i.e., an offline fashion. In contrast, we propose the first online token-pruning method that can be integrated seamlessly with current MLLM models for 3D question answering tasks, without additional training and with lower memory usage.Our key insight is to project each input frame into a shared voxel space using depth information and camera pose, identifying spatially-overlapped regions across frames and selectively pruning redundant image tokens before they enter the language model. Our method enables efficient online processing while reducing up to 50% of token usage. We apply this approach to Qwen2.5-VL-7B and Qwen3-VL-8B, demonstrating improved performance on the ScanQA, SQA3D, and OpenEQA-HM3D benchmarks.

Low-altitude Unmanned Aerial Vehicles (UAVs) often need to infer channel knowledge across a range of heights from only sparse observations collected at a few altitude layers. To address this challenge, this paper studies height-conditioned cross-height channel knowledge map (CKM) prediction for UAV-assisted communications in geometry-rich urban environments. We develop a geometry-aware conditional prediction framework that combines urban scene priors, sparse multi-altitude observations, and target-height descriptors to reconstruct dense CKMs at unobserved target heights. An uncertainty head is further introduced to characterize prediction confidence and to support cost-aware online UAV sensing under motion and safety constraints. Experiments on a layered aerial CKM benchmark show that the proposed Feature Pyramid Network (FPN)-Transformer achieves the best overall performance under both unseen-scene zero-shot and legacy patch-random protocols, reducing the Root Mean Square Error (RMSE) to 5.347dB and 1.111dB, respectively, compared with 6.937dB and 1.221dB for the strongest baseline 3D-RadioDiff. Moreover, after applying our unseen-scene few-shot adaptation, the RMSE further decreases from 5.347dB in zero-shot prediction to 3.518dB with 10-shot two-height support, while the uncertainty-guided cost-aware sensing policy improves active reconstruction from 6.94dB at initialization to 4.79dB at sensing budget 40, outperforming uncertainty-only sensing at 5.08dB and random aerial sampling at 5.84dB.

Autonomous driving systems benefit from high-definition (HD) maps that provide critical information about road infrastructure. The online construction of HD maps offers a scalable approach to generate local vectorized maps from onboard sensor observations. Existing methods commonly adopt bird's-eye-view (BEV) features as the intermediate scene representation, encoding the surrounding space with fixed-resolution dense grids. However, map elements are spatially sparse yet require fine-grained geometric localization, making uniformly allocated BEV representations redundant and less effective for vectorized map prediction. In this work, we propose GaussianMap, an online HD map construction framework that learns an adaptive Gaussian representation of the surrounding scene. This representation consists of a set of Gaussian primitives on the BEV plane, each encoding a flexible local region with geometric properties and a feature vector, allowing the model to allocate representational capacity to map-relevant regions. To generate such a representation from sensor observations, we introduce a feed-forward Gaussian encoder that progressively refines these primitives through Gaussian interaction modeling and multi-sensor feature aggregation. The refined Gaussian representation is then splatted into a BEV feature map and decoded into vectorized map predictions. Extensive experiments on nuScenes and Argoverse 2 datasets demonstrate that GaussianMap achieves state-of-the-art performance in both camera-only and camera-LiDAR fusion settings. Our code will be made publicly available.

We propose OTCache, a training-free framework for accelerating diffusion sampling via caching schedule prediction. Existing graph-based caching methods reduce redundant computation by optimizing shortest-path objectives, but rely on an additive independence assumption, which often breaks down in the low NFE regime. To address this issue, OTCache models caching schedules across inference budgets as a smooth evolution in policy space, inspired by Optimal Transport (OT). The framework consists of three stages: (1) obtaining a high-fidelity \textbf{reference schedule} using a graph-based caching method under a conservative budget; (2) performing a lightweight anchor search under an extreme low-budget setting via Optuna optimization with an end-to-end perceptual objective; and (3) predicting schedules for target budgets via quantile interpolation between the reference and anchor policies using continuous warping representations. Experiments on FLUX.1 [dev], Qwen-Image, and HunyuanVideo show that OTCache achieves 4.5x, 4.7x, and 3.66x acceleration, respectively, while consistently improving generation fidelity over state-of-the-art caching baselines. This work provides a new perspective on accelerating diffusion models through Optimal-Transport-inspired schedule modeling. Code:https://github.com/UnicomAI/OTCache

We present RenderFormer++, a scalable and physically grounded feed-forward neural rendering framework for global illumination in mesh scenes. Existing Transformer-based neural rendering methods such as RenderFormer achieve promising cross-scene generalization, but suffer from limited physical consistency and poor scalability due to the quadratic attention complexity of triangle-level tokenization. To address these issues, we introduce Physics-Informed Transport Guidance (PITG), which embeds rendering-equation inductive biases into the attention mechanism and enforces transport consistency loss, enabling physically consistent light transport modeling. We further propose Hierarchical Object-Centric Tokenization (HOCT), which aggregates triangle-level features into compact object-level tokens via cross-attention with learnable queries, substantially reducing computational and memory costs while preserving geometric and radiometric information. Extensive experiments demonstrate that RenderFormer++ achieves scalable, stable, and generalizable feed-forward global illumination rendering across complex large-scale scenes with improved physical accuracy and efficiency over prior neural rendering methods.

Precisely manipulating objects in a single photograph (translation, rotation, scaling) while obeying 3D physical constraints remains unsolved for diffusion-based editors. Current 2D methods lack spatial awareness and produce perspective violations. Forcing structural proxies into the latent space also disrupts variance homogeneity, and the resulting self-attention leakage leads to ghosting and background blur. The core difficulty is asymmetric: the relocated object must follow a rigid geometry, yet the uncovered background needs freedom to synthesize plausible content. We present GeoEdit, a training-free Lift-Manipulate-Render-Denoise pipeline that satisfies both constraints. We decouple scene and object in 3D, align them through point correspondence, and render a geometry-aligned proxy with a structural depth map. A Dual-Branch Denoising stage then refines this proxy: a video diffusion backbone preserves object identity, while 3D constraints are injected into the foreground within a narrow denoising window at matching noise variance (variance-homogeneous injection). The background denoises freely. Because the injected signal matches the native latent statistics, self-attention stays undisturbed. We also introduce GeoEditBench, a pose-aware benchmark covering object translation, object rotation, and camera movement with pose-aware evaluation metrics. Experiments confirm consistent gains in geometric accuracy, identity fidelity, and background quality. Our codes are available at https://github.com/Heey731/GeoEdit.

Classical universal approximation theorems establish the expressive power of sigmoidal multilayer perceptrons, but they do not prescribe how initial weights should encode the geometry of a data distribution. We propose S-GAI, a spectral geometry-aware initialization framework for one-hidden-layer sigmoidal MLPs. Starting from the constructive idea that sigmoid units can act as smooth half-space gates, we move from hand-specified planar geometry to class-wise spectral geometry estimated from image data. For each class, SVD provides a mean, principal directions, and spectral scales. An energy threshold selects the retained directions, and each retained direction is represented by two sigmoid gates. These class-specific gates form a shared hidden layer initialized directly from the training set. We also formulate a SVD-based subspace classifier as a non-neural geometric reference, which tests whether the estimated spectral class geometry is already discriminative before being embedded into the MLP. Experiments on MNIST, Fashion-MNIST, and a more challenging CIFAR-10 test show that the S-GAI-initialized MLP starts from a substantially more informative hidden state than Xavier initialization and reaches comparable final accuracy under full training. When the hidden layer is frozen, training only the output layer still gives stronger performance than frozen random gates, providing evidence that S-GAI effectively embeds class-wise spectral geometry into the MLP.

Mesh subdivision is a fundamental operation for converting coarse, editable meshes into high-resolution surfaces, with broad applications in digital asset creation. Classical rule-based schemes rely on fixed local refinement rules and often produce over-smoothed surfaces. Recent neural subdivision methods improve detail synthesis, but remain constrained by local modeling and exhibit limited generalizability. We present SubdivAR, a neural mesh subdivision framework based on our proposed Mesh Autoregressive Representation (MAR). MAR arranges meshes at different subdivision levels into an ordered scale sequence, reformulating subdivision as autoregressive next-scale prediction. To support this formulation, we introduce a Hybrid Topology-Aware Transformer that combines global semantic attention with topology-constrained local feature aggregation. SubdivAR adopts a next-scale coordinate prediction paradigm, regressing vertex offsets at each refinement stage to preserve subdivision topology while recovering fine-grained geometric details. To enable reliable learning, we construct FII-40K, a curated dataset of nearly 40,000 high-quality meshes with multi-level subdivision supervision. Experiments show that SubdivAR outperforms state-of-the-art baselines, reducing Hausdorff Distance and Chamfer Distance by 18.8% and 14.2%, respectively, and demonstrates strong robustness on complex open-surface geometries.

Neural combinatorial optimization (NCO) has shown that policies trained by reinforcement can construct strong solutions to NP-hard problems directly from raw instances. What such a policy actually learns, as opposed to what its decoder expresses, remains much less clear. We study this distinction on the vertex-guard Art Gallery Problem, the NP-hard task of choosing polygon vertices from which to observe an entire region. A pointer-network policy is trained from a coverage-aware reward over its own rollouts under the constraint we call geo-free inference: at test time it sees only vertex coordinates, with no visibility computation and no geometric oracle. The policy places guards economically but leaves a tail of under-covered polygons that widens far beyond the training range. To locate the cause, we freeze the trained encoder and read its embeddings with a small single-shot classifier, still geo-free at inference. The classifier closes most of the feasibility gap, in and out of distribution and at up to roughly five times the training range, cutting under-covered polygons by about an order of magnitude at an explicitly reported cost in guard count. We read this as evidence that the reinforcement-trained representation already encodes the geometry required for feasibility, and that residual failures reflect decoder calibration rather than missing knowledge. Probing a frozen encoder thus offers a practical way to ask what a neural combinatorial solver has internalized.

Table Structure Recognition (TSR) using a pointer network achieves impressive results by predicting HTML sequences while aligning tags to detected text (or cell) regions. However, our analysis reveals that when pointer networks fail, 79.6% of errors occur between spatially adjacent cells (Manhattan distance <= 2). Despite this, standard cross-entropy loss weights all negative candidates equally. In this work, we propose Geometry-Aware Pointer (GAP) Loss, which reweights the cross-entropy objective based on spatial proximity to ground truth. By applying inverse distance weighting, GAP focuses gradient flow where the model struggles most: immediate neighbors receive stronger gradients than distant cells. Our approach requires only a straightforward modification to the loss computation, maintaining the same model architecture with zero additional inference cost. Extensive experiments on PubTabNet and SynthTabNet demonstrate that GAP consistently reduces adjacent-cell errors, achieving new state-of-the-art performance. Our findings suggest that incorporating geometric inductive biases at the loss level provides a simple yet effective approach to robust TSR. Our code is available at https://github.com/teamreboott/GAP

Patient-specific 3D reconstruction of pelvic organ geometry from MRI is important for pelvic floor modeling and downstream patient-specific analysis. However, while previous studies have focused primarily on either image segmentation or downstream use of 3D models, the reconstruction of high-fidelity, high-quality geometries remains labor-intensive and poorly standardized. The study introduced a hybrid deformable shape modeling framework that integrates deep learning prediction with iterative optimization for the reconstruction of the bladder, uterus, and rectum. The framework consists of three core components: a geometry-aware multi-level deep learning architecture that preserves topological consistency of pelvic organs; a two-stage amortized optimization training strategy that balances global shape capture and local surface refinement; and a holistic synergy mechanism--where iterative optimization provides supervision for deep learning during the training phase, and during inference, deep learning rapidly predicts the global organ morphology, followed by iterative optimization to refine local surfaces and mesh quality. This framework demonstrated marked superiority in geometric fidelity than current mainstream deep learning-based organ reconstruction models. For individual anatomical structures, the reconstructed 3D geometries for the bladder, rectum, and uterus achieved significantly lower Chamfer Distance values and higher Dice Similarity Coefficient scores. In addition, while maintaining high computational efficiency, the proposed architecture yielded superior overall volumetric mesh quality. At the patient level, the framework achieved higher mean values for the 10 worst elements for both minSICN and minSIGE compared to traditional geometric post-processing algorithms.

Plate and shell structures are widely used in engineering, making rapid response prediction under varying geometries, materials, and loads highly desirable. However, conventional finite element methods require repeated modeling and solution, resulting in high computational costs. This study proposes a geometry-aware variational neural operator for Mindlin-Reissner plate problems, termed MR-GVNO. The method uses boundary point clouds to represent irregular geometries and employs separate encoders for spatially varying material fields, pressure loads, and scalar physical parameters. A cross-attention mechanism integrates these inputs with query point information to predict transverse deflections and rotations at arbitrary locations. MR-GVNO is trained without labeled solution data using a variational physics-informed loss derived from the discretized total potential energy. It directly processes irregular point clouds and allows different physical fields to be discretized independently, avoiding interpolation onto a common grid. Numerical experiments on single-hole, double-hole, and L-shaped plates demonstrate accurate response prediction under homogeneous and heterogeneous materials and uniform and random loads. The model also achieves millisecond-level full-field inference and favorable cross-geometry generalization.

Advanced surgical robotics has made robot-assisted endoscopic submucosal dissection (ESD) a promising approach for the en-bloc resection of large lesions, with the potential to reduce recurrence and improve long-term outcomes. However, the technical complexity and risk of complications in ESD demand stable and precise visual guidance to maintain an accurate dissection corridor and a safe tissue margin. Dense confidence fields provide an effective representation for this purpose by describing both the preferred dissection region and its spatial transition to surrounding tissue. However, reliable confidence field estimation remains challenging in dynamic endoscopic scenes due to smoke, specular highlights, tissue deformation, weak texture, and the thin geometric structure of the target region. To address these challenges, we formulate dissection guidance as a geometry-aware confidence field estimation problem and propose GeoCFNet, a geometry-aware confidence field network built on a pretrained DINOv3 backbone. GeoCFNet integrates a Token-Differentiated Fusion module to aggregate class-token context with dense patch representations, a SegFormer decoder for confidence regression, and Geometry-Aware Spatial Regularization (GASR) to preserve spatial coherence and local geometric transitions. Experimental results show that GeoCFNet achieves RMSE 0.0480, PSNR 27.1995, SSIM 0.3397, and CC 0.2466, indicating accurate and geometrically stable confidence field estimation for robot-assisted ESD guidance.

Indoor localization from wireless measurements remains challenging in large-scale deployments due to substantial variation in building geometry, the set of detectable access points (APs), and the heterogeneity of received signals. Existing learning-based methods often perform well only in limited settings and degrade under environmental shifts, making robust anchor-free localization across diverse indoor environments notoriously difficult. In this paper, we present OmniLoc, an environment-interactive foundation model for anchor-free user equipment localization across diverse indoor environments. To the best of our knowledge, OmniLoc is the first foundation-model-based approach built directly on wireless measurements for this task. OmniLoc is built on three key designs. First, a unified input tokenization module converts heterogeneous wireless measurements into a common representation that is more amenable to learning. Second, a geometry-aware Transformer performs AP-aware feature extraction by emphasizing dominant APs while aggregating complementary evidence from supporting APs. Third, a geometry-aware location estimation module conditions regression on geometric embeddings to produce geometrically consistent location predictions. We evaluate OmniLoc on both a large-scale in-house dataset and a public benchmark dataset. Results show that OmniLoc significantly outperforms existing methods, consistently improves existing backbones when its design components are integrated, and demonstrates strong generalization in cross-environment evaluations.

Traditional heuristic solvers for the 2D irregular nesting problem share a fundamental limitation: they are blind to polygon geometry, relying on guided brute-force to navigate the continuous placement space with minimal geometrical guidance. In this paper, we argue that Reinforcement Learning is uniquely positioned to overcome this bottleneck. By pairing an optimization policy with a geometry-aware neural encoder, an agent can automatically discover rich geometric priors directly from data, utilizing these learned intuitions to strategically guide exploration. To realize this, we introduce the Polygons Transformer (PoT), a novel architecture that encodes 2D continuous vector geometries while allowing cross-polygons attention. We couple this novel architecture with a Combinatorial Optimization Reinforcement Learning (CORL) training framework to find optimal solutions. To support this paradigm, we release an open-source training dataset derived from complex geographic contours alongside a dedicated evaluation benchmark. Our empirical validation demonstrates that our trained agent achieves area utilization performance highly competitive with Sparrow, the state-of-the-art heuristic solver, proving that reinforcement learning can successfully discover and exploit geometric awareness for precise spatial tasks.

As autonomous systems expand from capital-intensive robotaxis to cost-sensitive logistics, sensor configurations are increasingly optimized for coverage-per-cost. A prevalent sparse-view setup utilizes dual-fisheye cameras with a roof-mounted LiDAR, introducing severe geometric challenges: extreme radial distortion, minimal overlap, and misalignment between spherical projections and rectilinear grids. BEV fusion algorithms typically force image and point cloud modalities into unified Cartesian grids early in the pipeline, causing significant feature distortion and information loss for wide-view fisheye cameras. To address this, we propose a Geometry-Aware Hybrid Fusion (GA-HF) framework that explicitly accounts for fisheye geometry and BEV feature distortion, where fisheye features are lifted into a polar BEV grid via a Distortion-Aware Lift-Splat-Shoot (LSS) module to preserve native angular density, while LiDAR features are processed in native Cartesian space for metric fidelity of bounding box regression. To bridge these heterogeneous streams, we introduce a Dual-Attention Warping Correction module that applies spatial and channel attention to the warped camera features before fusion, explicitly suppressing artifacts in low-quality peripheral regions while enhancing high-quality semantic cues. GA-HF is evaluated on three benchmarks: KITTI-360, Dur360BEV, and Fisheye3DOD datasets. To the best of our knowledge, it is the first approach to explore LiDAR-fisheye camera fusion. On KITTI-360, GA-HF improves NDS by 4.2% over Cartesian baselines; on Dur360BEV, it surpasses both LiDAR-only and BEVFusion, while significantly reducing orientation error despite the geometric distortions; on Fisheye3DOD, it attains the highest detection score among all fusion methods.

Estimating local mean curvature at each point of a high-dimensional dataset is a key ingredient of geometry-aware machine learning algorithms, such as the Mean Curvature Boundary Points (MCBP) method. The naive implementation of this computation, based on a local shape operator approximated from k-nearest neighbor patches, involves an explicit construction of a matrix $H$ whose trace form yields an $O(m^4)$ cost per point, rendering the approach intractable for datasets with more than a few dozen features. This paper introduces two complementary contributions that together reduce this cost by several orders of magnitude. The first contribution is an exact algebraic identity. This identity, derived from the orthogonality of the eigenvectors of the covariance matrix and the cyclicity of the trace operator, eliminates $H$ entirely and reduces the per-point cost to $O(m^2)$ after the eigendecomposition. The second contribution addresses the remaining $O(m^3)$ bottleneck of the full eigendecomposition. Since the local covariance matrix has rank at most $k-1 \ll m$, we replace it with a truncated SVD of the $k \times m$ centered data matrix, an $O(k^2 m)$ operation, and derive an analytical approximation for the contribution of the null-space eigenvectors based on the expected value of their outer product under the Haar measure. The resulting estimator has total cost $O(k^2 m + k m p^2)$, where $p = k-1$. Experiments on real-world datasets confirm speedups of 50 to 300 times relative to the original implementation, with negligible loss when the fast estimator is used to replace the original version. By providing a scalable and data-driven estimate of local curvature, the proposed method establishes curvature as a practical geometric feature for a broad range of machine learning tasks, from classical to modern deep learning pipelines.

Dataset condensation aims to construct compact datasets from real data via synthesis or selection. However, existing approaches are ill-suited for diffusion model training: synthetic data generation often yields low-fidelity samples unsuitable for authentic modeling, while real subset selection typically fails to preserve the distributional geometry required by diffusion likelihood objectives. To address this, we propose to reformulate real subset selection as a geometry-aware distribution alignment problem. By incorporating one-sided partial optimal transport, our method selectively aligns a compact subset with the full data distribution while allowing unmatched mass in low-density regions, ensuring the preserved geometric structure necessary for effective diffusion model training. To further ensure distributional fidelity, we complement geometric alignment with lightweight feature-statistics and semantic consistency regularization. An efficient two-stage discrete optimization strategy is proposed to achieve this alignment objective. Extensive experiments across diffusion variants, subset sizes, image resolutions, and training rounds show that our method achieves superior fidelity and distributional coverage in diffusion model training. Codes are available at https://github.com/2018cx/GADC.

While Proximal Policy Optimization (PPO) demonstrates strong performance in stationary settings, we show that its standard optimization paradigm struggles in continual and non-stationary environments. The failure does not stem from insufficient model capacity or overly restrictive clipping. Instead, PPO performs persistent, directionally inefficient local updates, which indicates a lack of geometry-aware guidance for accumulating meaningful behavioral change and ultimately hindering transitions toward new behavior patterns. Although divergence-based regularization introduces partial geometric awareness, its monotonically increasing penalties implicitly discourage large policy deviations, even when such shifts are necessary for effective adaptation. To address this limitation, we propose Gaussian Trust Region Policy Optimization (GTR), which reshapes the trust region using a Gaussian kernel. The resulting constraint is bounded and non-monotonic, providing strong local stability while progressively relaxing under sustained high-advantage updates. To further improve robustness, we introduce a Mixture Gaussian Anchor that adapts to recent policy trajectories, reducing variance induced by stale references. GTR is architecture-agnostic and achieves strong performance across games, simulated robotic control, open-world exploration, and language model post-training. These results demonstrate that geometry-aware trust-region design can be a promising direction for robust reinforcement learning in complex non-stationary environments. Our code is available at https://anonymous.4open.science/r/GTR_demo/README.md.

Video world models aim to simulate controllable visual environments, but long-horizon rollouts depend on what the model remembers after observations leave its native context window. Explicit memories retain frames or online 3D reconstructions, which can suffer from heuristic retrieval errors, redundant appearance storage, or reconstruction artifacts. Implicit memories compress history into a compact state, but existing designs are not explicitly constrained to encode cross-view scene geometry. We propose GIM-World, a geometry-aware implicit memory framework for video world models. A lightweight transformer encoder compresses variable-length history into fixed-size memory tokens, a camera-queryable geometry head distills 3D scene structure from a frozen foundation model into the memory during training, and an information-guided pruning rule keeps encoding cost bounded as history grows. The geometry teacher is discarded at inference, leaving a lightweight memory module. Experiments on MIND show that GIM-World better preserves long-horizon geometric and visual consistency than both explicit- and implicit-memory baselines.

Discrete visual tokens should provide a compact representation for both token-based world modeling and planning in autonomous driving. However, most tokenizers are inherited from image generation and are optimized mainly for pixel reconstruction, which may leave a gap between what is easy to generate and what is useful to decode for driving decisions. We present a representation-guided and geometry-enhanced tokenizer that learns discrete tokens under joint supervision. The tokenizer aligns its discrete bottleneck with a frozen DINO feature space through feature decoding, while preserving appearance via RGB reconstruction with perceptual and adversarial losses. To inject geometric state-related cues, we add adjacent-frame depth and relative-pose supervision during training and stabilize joint objectives with multi-codebook quantization. We evaluate the same learned tokens with a lightweight planning readout and a GPT-style next-token world model. Experiments on NAVSIM show improved reconstruction fidelity and representation consistency, competitive planning performance under a fixed decoder, and better generative quality under matched settings.

Geometry-aware generative models and novel view synthesis approaches have shown strong potential in visual fidelity and consistency. In parallel, equivariant representation learning has emerged as a powerful framework for constructing latent spaces where analytically known group transformations could act directly, capturing geometric structure in data and enhancing both interpretability and generalization in novel view synthesis. However, we identify that existing approaches often suffer from latent misalignment, a discrepancy between the intended group action and the actually required transformations in the latent space. Consequently, the learned latents often fail to consistently preserve the equivariant relations imposed by the underlying group symmetry. To address this, we propose Residual Latent Flow, a flow-based framework that corrects the misaligned latents, thereby improving compliance with the underlying equivariance relation. Our comprehensive experiments show that our method significantly reduces latent misalignment and improves novel view synthesis quality, under rotation groups SO(n).

3D object grounding localizes referred objects in a 3D scene from natural language. Unified instance-centric 3D-LLMs aim to solve grounding together with dialog, QA, and captioning, yet many rely on a single pointer-style grounding decision that compresses a relational instruction into one selection. This is brittle for fine-grained queries where multiple same-class candidates must be ruled out by context objects and spatial relations. We propose Structured Spatial Reasoning 3D-LLM (SSR3D-LLM), a structured grounding interface for unified 3D-LLMs. Given fixed Mask3D object proposals, the LLM writes a sequence of latent spatial reasoning steps and memory tokens from the query, and a geometry-aware scorer reads these latent steps in order to refine candidate rankings step by step with step-length masking. The latent steps are learned from standard benchmark target supervision with auxiliary referential-cue supervision during training, while inference uses only the input query and Mask3D proposals. Across ReferIt3D, ScanRefer, and Multi3DRef, SSR3D-LLM achieves the strongest results among unified 3D-LLM baselines, with substantial gains over the single-pointer QPG baseline on fine-grained grounding and consistent improvements over prior unified 3D-LLMs, while preserving the default language-task route.

Generating collision-free and smooth motions remains a central challenge in robotic manipulation, particularly in cluttered environments and narrow passages where feasible regions are highly constrained and fragmented. We propose a trajectory optimization framework that performs geometry-aware updates directly in function space using natural functional gradients. The method optimizes a Gaussian-smoothed surrogate objective that regularizes the optimization landscape through smooth trajectory perturbations while preserving trajectory-level structure. Because the updates are defined intrinsically in function space, trajectory regularity can be controlled independently of a particular time discretization. We derive a practical Monte-Carlo estimator of the natural functional gradient that requires only black-box trajectory evaluations, making the method applicable when analytic gradients are unavailable or unreliable due to collision checking and contact-rich simulation. Experiments on constrained robotic manipulation tasks demonstrate that the proposed method improves trajectory feasibility and produces smoother motions than representative planning and trajectory optimization baselines in environments with narrow geometric clearances. Additional results, videos, and implementation details are available at the project page: https://kisangpark.github.io/natural-functional-gradient/

Automotive crashworthiness optimization remains a safety-critical challenge, requiring the management of large-scale nonlinear structural deformations and energy dissipation through iterative, high-fidelity simulations. While traditional finite element solvers are computationally prohibitive, emerging operator learning frameworks provide rapid surrogate predictions; however, applying them to industrial-scale crash analysis, where complex geometry, contact nonlinearities, and rapidly evolving transient deformation coexist, remains an open challenge. In this paper, we demonstrate that the GeoTransolver framework provides a viable solution for accurate, high-fidelity crash dynamics prediction at industrial scale. Benchmarked on complex bumper beam and full-vehicle crash datasets, GeoTransolver captures multi-scale geometric context and accurately resolves plastic deformation patterns as well as acceleration profiles at critical occupant locations. Beyond the architecture itself, we propose and systematically evaluate a suite of temporal prediction recipes, including one-shot, time-conditional, and autoregressive rollout strategies, demonstrating that the one-shot approach achieves state-of-the-art accuracy with significantly reduced training overhead and inference latency. As a secondary contribution, we introduce a Fast Low-rank Attention Routing Engine (FLARE)-based modification to the GeoTransolver attention backbone that reduces memory overhead by approximately 2x while further improving predictive accuracy for O(N) long-range, high-frequency transients, preserving the geometry-aware cross-attention strengths of the base framework. Our results highlight the practical viability of geometry-aware operator learning for high-fidelity surrogate modeling of complex, safety-critical automotive dynamics.

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.

Steering a language model - intervening on its internal activations to change downstream behaviour - has recently expanded beyond linear interpolation to nonlinear methods such as angular and kernelized steering, which define intervention transformations without learning an explicit geometry over paths in activation space. Freshly introduced geometry-aware manifold methods do learn such a geometry, but require labelled class centroids together with prescribed cyclic or sequential structure. These assumptions restrict where manifold steering can be applied, since existing constructions require labelled centroids and compatible boundary conditions. We recast manifold steering more broadly as \textbf{Riemannian geodesic computation} on activation space, recovering linear and labelled-spline steering as geodesics under particular choices of metric. A principled metric within this framework is the output-space Hellinger distance pulled back to activations; we approximate this with a learned encoder trained on output distances over a small concept-token schema - no per-prompt labels, no topology prior, and no per-task curve fitting. Empirically, the method reliably drives the model onto the target class across all tasks in a standard four-task language-model arithmetic benchmark, while following more behaviourally natural trajectories than baselines on smaller output spaces. We thereby provide a unified Riemannian framework for manifold steering together with a schema-supervised, label-free instantiation that operates without labelled centroids or prescribed boundary conditions.

We propose Drift-Resistant Navigation World Model, a generative model that mitigates both perceptual drift and geometric drift in conventional rollout-based navigation world models. Existing methods recursively feed generated content into subsequent steps, causing noise accumulation and degraded predictions, i.e., perceptual drift. Meanwhile, their predictions often deviate from the agent's motion, resulting in geometry drift. We address both types of drift by redesigning world-model prediction as an anchor-guided rollout. Instead of rolling out every frame sequentially, we first predict sparse future anchors that serve as stable long-range targets, and then generate intermediate frames within each chunk conditioned on both past context and future anchors. Importantly, these sparse anchors also provide geometric constraints, supported by bidirectional epipolar geometry, to localize where corresponding content should appear in the intermediate frames. Experiments on four benchmarks demonstrate consistent improvements over strong baselines in long-horizon visual quality, geometric consistency, and multi-view coherence. These gains further translate into improved downstream planning performance under the same planners, highlighting the importance of drift-resistant, geometry-aware prediction for reliable navigation world models.

As wearable and mobile devices become increasingly embedded in daily life, they offer a practical way to continuously sense human motion in the wild. But inertial signals are highly dependent on the sensing setup, including body location, mounting position, sensor orientation, device hardware, and sampling protocol. This setup dependence makes it difficult to learn motion representations that transfer across devices and datasets, and limits the broader use of wearable IMUs beyond closed-set recognition. We introduce AnyMo, a geometry-aware framework for setup-agnostic human motion modeling. AnyMo uses physics-grounded IMU simulation over dense body-surface placements to generate diverse and plausible synthetic signals, pre-trains a graph encoder from paired synthetic placement views and masked partial observations, tokenizes multi-position IMU into full-body motion tokens, and aligns these tokens with an LLM for motion-language understanding. We evaluate AnyMo on three complementary tasks: zero-shot activity recognition across 14 unseen downstream datasets, cross-modal retrieval, and wearable IMU motion captioning, where it improves average Accuracy/F1/R@2 by 11.7%/11.6%/22.6% on HAR, increases zero-shot IMU-to-text and text-to-IMU retrieval MRR by 15.9% and 28.6%, respectively, and improves zero-shot captioning BERT-F1 by 18.8%. These results support AnyMo as a generalist model for wearable motion understanding in the wild. Project page: https://baiyuchen.com/project/AnyMo.

Spatio-temporal reasoning in vision-language models requires visual representations that preserve physical geometry rather than merely semantic appearance. Recent multimodal models incorporate geometric information through structural branches, 3D-aware supervision, reasoning-stage fusion, or long-horizon memory. While these approaches demonstrate the importance of geometry for spatial intelligence, they typically treat geometric cues as a shared signal across all visual tokens. We note that this overlooks a finer-grained challenge: different visual tokens require different geometric evidence depending on their spatial roles. To address this limitation, we introduce GeoWeaver, a pre-reasoning geometric grounding framework that treats geometry as a representational prerequisite for spatio-temporal reasoning. GeoWeaver constructs a multi-level geometry bank from a frozen geometry encoder and performs token-adaptive geometric evidence allocation, enabling each visual token to retrieve the most relevant geometric abstractions. The selected evidence is incorporated into visual tokens via a residual grounding operation prior to language modeling, yielding geometry-grounded representations for downstream reasoning. Extensive evaluations on spatial reasoning benchmarks demonstrate that GeoWeaver consistently enhances geometry-aware reasoning while retaining general multimodal capabilities. This indicates that geometric information yields the greatest benefit not as a late-fusion auxiliary signal but as a fundamental prerequisite that shapes the representational foundation on which large language models perform reasoning. All source code and models will be released at https://github.com/yahooo-m/GeoWeaver .

Correlation matrices are fundamental summaries of functional brain networks, yet standard analyses often treat entries independently, ignoring the curved geometry of correlation space. Existing geometric methods frequently lack closed-form operations or depend on arbitrary region ordering, limiting scalability. We introduce a scalable geometric framework with two components: (i) the Off-log metric, a smooth transformation mapping correlation matrices to symmetric zero-diagonal matrices. This enables closed-form expressions for distances, Frechet means, and linear models, allowing standard statistical modeling without complex manifold optimization. (ii) Grassmannian subspace discrimination, which compares subjects via principal-angle distances between eigenvector subspaces, resolving inherent sign and basis ambiguities. Both components integrate into standard machine-learning workflows for inference, regression, and classification. Validated across two clinical cohorts (Parkinson's and psychosis) and three ageing fMRI datasets, the Off-log metric increased sensitivity in permutation tests and matched or exceeded Riemannian and Euclidean baselines in classification. Brain-age prediction performance was comparable, with Riemannian metrics excelling in two of three cohorts. The Grassmannian method consistently outperformed Euclidean baselines, highlighting disease-relevant networks. Overall, geometry-aware representations improve sensitivity and predictive performance while remaining straightforward to deploy at scale.

Post-training has become central to improving reasoning and alignment in large language models, where critic-free models enable scalable learning from model-generated outputs but lack principled mechanisms to distinguish informative from noisy signals. Recent approaches leverage response-level measures as uncertainty signals to regulate group-based optimization methods such as GRPO. Yet their empirical success remains unstable and unclear in how they influence optimization dynamics. In this paper, we provide, to our knowledge, the first principled formulation that interprets uncertainty signals as mechanisms for characterizing and regulating gradient variance and learning signal quality. Based on both empirical and theoretical analysis, we identify two critical gaps of current entropy-based estimators: The anisotropic gap and The calibration gap. Motivated by this analysis, we propose Geometric-aware Calibrated Policy Optimization (GCPO), a novel framework integrating geometry-aware measures to capture semantic disagreement with reward-based calibration to align uncertainty with learning signal strength. Experiments on multiple benchmarks show that our approach more faithfully tracks gradient variability and consistently improves post-training performance. Our results highlight the importance of designing uncertainty signals that are aligned with optimization dynamics, offering a principled perspective for robust post-training.

Existing Gaussian avatar methods typically parameterize geometry on a body-template surface, which entangles the avatar's representation space with the template's deformation space and limits the capture of layered, off-body, and non-rigid clothing geometry. We present PiG-Avatar, which addresses this limitation by using the parametric body model solely for kinematic transport, while representing the avatar as Gaussians anchored in a volumetric canonical space governed by a continuous neural field. This decouples representation from template topology, avoiding the geometric constraints of surface-based parameterizations. Kinematic coherence is maintained through 3D barycentric anchor transport, which guides motion without constraining geometry and allows anchors to deviate freely from the template surface, yielding dense, stable temporal surface correspondences by construction. To make this unconstrained formulation tractable, we introduce dual-level spatially coherent optimization, combining Sobolev-preconditioned neural-field updates with a novel KNN-based preconditioning of canonical anchor geometry. Together, these mechanisms induce an emergent self-organization of anchor density: anchors migrate toward regions of high curvature, appearance variation, and non-coherent motion without explicit heuristics. As a result, complex clothing geometry and layered surfaces emerge as natural, high-fidelity outputs. This single representation further supports hierarchical reconstruction across multiple levels of detail, with coarse-level supervision propagating to finer levels through the shared field and coupled anchor graph. On established benchmarks featuring subjects with complex clothing and challenging non-rigid motion, PiG-Avatar achieves state-of-the-art rendering quality, generalizes robustly to imperfect body model initialization, and renders in real time across all detail levels.