Geometric Representation Learning

Contrastive learning is increasingly moving toward settings with three or more modalities instead of image-text pairs. Yet, extending models from pairwise to higher-order multimodal alignment can introduce optimization and representation challenges. We identify encoder Jacobian conditioning as a key factor in trimodal contrastive learning: poorly conditioned encoders exhibit collapsing or amplified singular-value spectra, leading to exploding Jacobian condition numbers and degraded multimodal alignment. We introduce geometry-preserving encoders (GPEs) by directly conditioning the Jacobian through regularization and demonstrating that simple modifications like LeakyReLU activations and residual paths recover these geometric benefits. Across a synthetic benchmark and four real-world datasets including missing modalities, improving Jacobian conditioning boosts retrieval and linear probe performance across multiple contrastive objectives, whereas expressive objectives yield little benefit in linear probes. More broadly, our results show that multimodal contrastive learning depends not only on objective expressivity, but also on the geometric and optimization properties of the underlying encoders.

Accurate prediction of crystal properties remains a key challenge in computational materials science. While graph neural networks (GNNs) such as CGCNN, MEGNet, ALIGNN, and SchNet have shown strong performance, they primarily represent crystals at the atomic level and implicitly learn local chemical environments through message passing. However, many material properties are governed by coordination polyhedra, the fundamental structural units formed by atoms and their neighboring atoms. To address this limitation, we propose the Coordination Polyhedron Graph Network (CPGN), a multi-scale GNN that jointly learns atomic, bond, and coordination-polyhedron representations. CPGN constructs three coupled graphs: an atom graph encoding elemental and bonding information, a line graph capturing angular interactions, and a coordination polyhedron graph describing Voronoi-derived local environments through corner-, edge-, and face-sharing relationships. Physically meaningful geometric descriptors are incorporated for each polyhedron, while an interleaved message-passing mechanism with bidirectional cross-attention enables effective information exchange across structural levels. Extensive evaluations on the Materials Project, JARVIS-DFT, and QM9 benchmark datasets demonstrate that CPGN outperforms existing state-of-the-art GNN models. It achieves a formation-energy MAE of 0.060 eV/atom and a band-gap MAE of 0.292 eV on the Materials Project, while providing competitive multi-property prediction on JARVIS-DFT and superior HOMO prediction on QM9. The results highlight that explicit modeling of coordination polyhedra improves crystal representation learning and enables accurate, physically interpretable prediction of material properties.

3D Gaussian Splatting (3DGS) captures scenes by coupling explicit geometry (position, covariance) with view-dependent photometry (Spherical Harmonics). However, building $\mathrm{SE}(3)$-equivariant architectures on these primitives presents a fundamental representation bottleneck. Color has been treated as a signal rather than a geometric entity, making it nontrivial to unify symmetry across geometry and appearance as the camera frame changes. While translations are handled by relative coordinates, rotations act heterogeneously across attributes: $μ\mapsto Rμ$, $Σ\mapsto RΣR^\top$, and $f_\ell\mapsto D^\ell(R)f_\ell$. This mismatch complicates strict equivariance, leading existing methods to either discard or flatten SH coefficients, thereby breaking symmetry. We propose a unified solution rooted in representation theory: for SH degrees $\ell\le2$, photometry is algebraically isomorphic to a rank-2 geometric tensor. We prove that the Wigner-$D$ action on these SH coefficients can be exactly reformulated as the conjugation action on $3\times3$ matrices. Leveraging this, we introduce the Unified Matrix Embedding, a lifting that maps all Gaussian attributes into a unified carrier space, $\mathfrak{gl}(3)$. Building on the "Color-as-Geometry" formulation, we present E3DGS, a rigid-body ($\mathrm{SE}(3)$) equivariant architecture that processes 3D Gaussians without Clebsch-Gordan tensor products. Evaluations on object vision and action-conditioned Gaussian world modeling demonstrate that our unified approach yields strong robustness under camera-frame changes and improved data efficiency.

CAD-to-image alignment aims to estimate an object's 9D pose (rotation, translation, and anisotropic scale) from a single RGB image, enabling applications in robotics and augmented reality. Recent zero-shot methods use visual foundation models to match image regions to CAD models, yet typically their correspondences are appearance-driven and degrade under occlusion or sim-to-real domain shift. To address these limitations, we introduce SUFLECA (Scaling Up Feature LEarning for CAD Alignment), a weakly-supervised framework for zero-shot CAD alignment with two key contributions. First, SUFLECA scales up geometry-grounded feature learning from pretrained visual representations through Normalized Object Coordinates (NOCs) supervision on 674K images spanning 12 real and synthetic datasets, learning compact geometry-aware features that generalize across domains. Second, we propose a geometrically consistent matching algorithm that establishes reliable one-to-one CAD-to-image correspondences. Together, these contributions enable accurate, sub-second alignment per object instance without iterative pose refinement. On ScanNet25k, SUFLECA achieves 33.4%/42.3% category/instance accuracy, outperforming, with a smaller computational footprint, the strongest zero-shot baseline by 10.3/12.2 percentage points and, for the first time on this benchmark, even surpassing fully supervised methods. Code is available at: https://github.com/snt-arg/SUFLECA

In this paper, we ask whether vision foundation models construct representations that reflect the intrinsic properties of 3D Euclidean space. Unlike previous works that probe 3D awareness of vision features by regressing image-centric quantities such as depth or normals, we investigate the relation between the structure of the space of visual features and the group of Euclidean transformations $SE(3)$. We propose a set of probes to evaluate this relation from both topological and geometric perspectives: a mutual neighborhood metric that measures the alignment between feature neighborhoods and spatial topology, and a Poincaré Adapter to test the linear accessibility of the geometry of camera motion from latent displacements in static scenes. We show that self-supervised vision models, which, in principle, have not been trained with direct 3D supervision or active agency, possess latent subspaces that are remarkably strongly correlated with three-dimensional Euclidean space, when probed correctly. Building on this insight we propose a new class of "Latent-Space Navigation" techniques that perform visual odometry and localization purely in the latent space, bypassing the need for explicit 3D reconstruction.

In this work, we introduce CoCo, a loss function aimed at learning normalized and well-structured representations. The proposed loss encourages intra-class collapse and inter-class contrast while preserving sufficient flexibility for neural networks to approximate geometrically optimal embeddings with large angular separation between classes. We provide a theoretical analysis positioning CoCo with respect to related objectives such as dot regression and cross-entropy, showing that the new proposed loss benefits from closer initialization to the optimal configuration, more informative gradients, and stronger incentives for class-wise representation collapse. Extensive experiments on diverse tabular datasets from the OpenML-CC18 benchmark show that CoCo achieves competitive performance with state-of-the-art methods, including kernel SVM, Random Forest, dot regression, and cross-entropy-based neural networks. In addition, both theoretical arguments and empirical analyses demonstrate that the proposal promotes tighter class clustering and faster convergence. These results highlight CoCo loss as an effective objective for learning discriminative representations while maintaining competitive predictive performance.

Existing generative models learn data distributions in flat Euclidean space. However, most data in our real world are manifolds embedded in high dimensional Euclidean space. Therefore, we propose an intrinsic-geometry-based generative adversarial network (IG-GAN) for data generation in the field of aerodynamics. The generator of the IG-GAN represents aerodynamic data as a piecewise smooth manifold constructed by Bézier surfaces, and the generator tries to learn the coefficients of each Bézier surface to further combine multiple Bézier surfaces into a smooth manifold automatically. The discriminator in the IG-GAN is a radial-basis-function based discriminator (RBF-D). Experimental results show that IG-GAN achieves lower predicted Mean Squared Errors (MSEs) than those of three baselines. Specifically, on the Burgers' equation dataset, IG-GAN reduces the predicted MSE of velocity u by 97.41% compared with state of the art SSL-Transformer. Additionally, on the ONERA M6 aircraft dataset, IG-GAN reduces the overall MSE of nine aerodynamic coefficients by 82.95% compared with SSL-Transformer.

Category-level object pose estimation is a crucial yet challenging task in both academia and industry, and has achieved remarkable success by leveraging keypoint-based correspondence paradigms. However, most existing methods increasingly rely on stronger feature learning while overlooking whether the established correspondences are geometrically stable across diverse perturbations. This often results in fragile pose recovery under intra-class shape variations and occlusions. To tackle this challenge, we develop a novel Triangle-Invariant Geometric Consistency Learning for Category-Level Object Pose Estimation (TriCons-Pose) to anchor stable keypoints and aggregate pose-invariant cues, yielding reliable canonical mapping and accurate pose estimation. Specifically, a Structure-Consistent Keypoint Detector (SCKD) is designed to identify robust keypoints by enforcing cross-view structural consistency via normalized pairwise distance matching. Moreover, we propose a Pose-Invariant Geometric Aggregator (PIGA) to augment keypoint representations by injecting triangle-based pose-invariant descriptors into a local-to-global attention mechanism. The proposed framework is optimized using standard objective functions while incorporating an additional geometry consistency loss. Extensive experiments on REAL275, CAMERA25, and HouseCat6D datasets demonstrate the effectiveness of the proposed approach.

Deformable shape representations have proven to be robust complements to texture features in cardiac image classification, offering geometric priors that are invariant to imaging artifacts and intensity variations. However, existing deep networks perform simple concatenation to combine these distinct feature representations, which neither fully exploits their complementary nature nor learns cross-modal feature dependencies. Furthermore, this results in uniform attention across all timepoints; hence ignoring the varying diagnostic importance across the cardiac phases. In this paper, we propose a novel cardiac video classification model that, for the first time, learns temporal features in an integrated space of deformable shape and image texture representations. In particular, we design a bi-directional cross-attention in the latent space to fuse latent deformable shape and image features, allowing each modality to adaptively weight the other based on spatio-temporal correspondence. In contrast to current methods that apply uniform weighting across all the cardiac phases, our approach learns to dynamically adjust the contributions of shape and texture representations, derived from images, over time. We demonstrate state-of-the-art classification performance on a cine cardiac magnetic resonance (CMR) video dataset, achieving improved interpretability from attention mechanisms that identify diagnostically critical cardiac phases and modality contributions.

Understanding the geometric structure of pre-trained language model embeddings matters for interpretability and safety. We ask whether sentence-level classification signal lives in the Riemannian geometry of contextual token embeddings, and probe it by extracting per-token pullback metrics from a learned encoder's analytical Jacobian and aggregating them with the Fréchet mean on the symmetric positive definite (SPD) manifold; we call this procedure Riemannian Mean Pooling (RMP). Across three datasets with non-trivial linguistic structure (CoLA, CREAK, RTE), RMP outperforms Euclidean mean pooling, while on FEVER-Symmetric, a benchmark constructed to remove annotation-driven lexical artifacts, the method correctly stays at chance. Ablations show that a randomly initialised encoder combined with Fréchet aggregation already beats Euclidean pooling on two of the three signal-bearing datasets, localising the source of the gain to the geometric aggregation rather than to learned manifold structure; the trained encoder contributes additional signal specifically on CREAK, the most knowledge-heavy of the three signal-bearing datasets.

World Action Models (WAMs) have shown strong potential for robotic manipulation by jointly modeling visual future dynamics and executable action sequences. However, existing video-action co-training methods primarily optimize appearance-oriented video latents, which may insufficiently capture the temporally evolving geometry required for precise manipulation. We propose MECo-WAM, a Multi-Expert Co-Training World Action Model that injects action-relevant 4D geometric priors into video-action representations while preserving the original lightweight inference graph. During training, MECo-WAM combines video and action experts with a lightweight 4D expert supervised by relational targets from a frozen VGGT encoder. Asymmetric expert visibility prevents non-causal shortcuts from auxiliary geometry to action generation. To transfer geometric knowledge into the deployed video-action pathway, we introduce decayed 4D read-mask attention, which provides restricted current-frame geometric guidance early in training and progressively removes this dependency. We further propose action-aware temporal geometric distillation, which aligns within-frame geometric relations and their temporal evolution while emphasizing visual regions most relevant to robot actions. At deployment, all auxiliary 4D components are removed. Experiments on LIBERO (98.2%), RoboTwin 2.0 (92.6%), and challenging real-world manipulation tasks show that MECo-WAM improves manipulation performance without increasing inference cost.

Learning motion latents for robotic manipulation heavily relies on extracting motion patterns from visual sequences, yet effective action abstractions require understanding three-dimensional geometric transformations. Here, we introduce GeoMoLa (Geometry-Aware Motion Latents), which learns discrete motion latent codes by predicting how point clouds evolve during manipulation rather than reconstructing visual observations. This four-dimensional objective -- spatial geometry changing through time -- forces latent representations to encode actual physical motion rather than appearance patterns. GeoMoLa achieves state-of-the-art performance using only single-view RGB-D input, while existing methods require multi-view reconstruction, succeeding across diverse manipulation benchmarks. Our ablations reveal that geometric prediction is the key to driving performance, quantitatively validating that manipulation depends on spatial understanding. Furthermore, the learned codes exhibit effective motion abstraction: applying them to novel scenes produces physically consistent transformations regardless of visual context. Our real-world experiments also confirm this robustness capability, achieving robust manipulation with minimal demonstrations in cluttered environments where geometric reasoning determines success. Thus, we demonstrate that effective motion latents for robot control can better emerge from understanding motion through its three-dimensional effects rather than pixel-level patterns.

Recent work showed that language models represent character counts on curved 1D manifolds, with attention heads performing geometric transformations to enable computation. We test whether this generalizes across four ordinal tasks (bracket depth, indentation, table position, numeric magnitude) in Gemma-2-2B, Gemma-2-9B, and Qwen3-4B. We find that 1D manifolds with place-cell feature tiling emerge for tasks where the ordinal variable is locally computable from token identity, while tasks requiring cross-position integration or semantic extraction produce higher-dimensional or incoherent representations. Geometric computation is architecture-dependent: Qwen3-4B shows substantially stronger twisting than Gemma models for indentation, and its twisters preserve ordinal order, unlike its numeric twisters. Activation patching confirms that the identified manifold subspaces concentrate task-relevant information, with manifold-direction ablation causing dramatically larger probe accuracy drops than random-direction controls.

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.

Descriptor-free visual localization eliminates high-dimensional descriptor storage, preserves scene privacy, and simplifies map maintenance, yet its accuracy still lags far behind descriptor-based pipelines. We identify this gap to insufficient geometric discriminability in geometry-only matching. Without visual appearance, current methods underutilize local geometry cues, lack the global context among keypoints, and overfit to a single keypoint detector. We further observe that descriptor-free matching naturally enables multi-detector training, as heterogeneous keypoints can be optimized in a shared geometry-only space without aligning descriptor spaces. Building on these insights, we propose GeoMix, a descriptor-free 2D-3D matching framework that strengthens geometric discriminability at three levels. Locally, directional and distance-aware embeddings enrich neighborhood aggregation with fine-grained spatial structure. Globally, learnable context nodes aggregate and redistribute scene-wide information via cross-attention to resolve ambiguities beyond local receptive fields. At the training level, Mix-Training exploits this detector-agnostic geometry space to learn representations across multiple keypoint detectors. Extensive experiments on MegaDepth, Cambridge Landmarks, 7Scenes, and Aachen Day-Night show that GeoMix sets a new state of the art among descriptor-free methods, reducing 75th-percentile rotation error by 89% and translation error by up to 90% over the previous best, while generalizing zero-shot to unseen detectors and narrowing the gap to descriptor-based pipelines. Code is available at $\href{https://github.com/YejunZhang/Geomix}{\text{this links}}$.

We introduce a controlled subspace intervention framework to investigate how self-supervised Vision Transformers (ViTs) encode dense geometric information. While linear probing is widely used to assess geometric representations, it treats features as a black box, failing to disentangle the underlying topology. To address this issue, we decompose the weights of converged linear probes to isolate the low-rank subspaces containing explicit geometric signals using Singular Value Decomposition (SVD). Our perspective yields three key insights: (1) Pre-training objectives determine how features are encoded. DINOv2 aligns spatial features for efficient linear extraction, while Masked Autoencoders (MAE) tend to disperse these signals, requiring a broader spatial context. (2) Explicit geometric representations are highly compressible, suggesting dense predictive heads could potentially be constrained to low-rank subspaces with minimal performance loss. (3) The layer-wise task affinity suggests that geometric precision peaks at intermediate layers before yielding to semantic abstraction in the final layers. By connecting internal encoding mechanics with downstream performance, these findings provide a basis for effective feature selection and lightweight decoder design. The source code is available at https://github.com/Zhou-Weichen/Geosubprobe.

Recurrent representations are trajectories, but representation geometry is often measured from static snapshots. We develop finite-lag operator geometry for recurrent hidden states from observed source-successor pairs $(X_t,X_{t+Δ})$. The primitive is the conditional transport law $Q_Δ(dy\mid x)$, estimated by a dense Gaussian source-smoothing operator. From this directed finite-lag law we derive a source-centered transport tensor $G_Δ$, which decomposes exactly into conditional spread and coherent displacement, and an antisymmetric coordinate circulation $W_Δ^ρ$, which summarizes directed lagged flow. We prove affine covariance with explicit metric dependence of scalar summaries, dense estimator stability on bounded trajectory clouds, and a finite-lag separation result showing that source-centered transport detects deterministic recurrent motion not recorded by infinitesimal carre-du-champ geometry. A linear-Gaussian closed form calibrates the quantities in terms of the update $A_Δ$, source covariance, and innovation covariance. Controlled experiments validate the decomposition, circulation, covariance, and stability predictions. In performance matched repeat-copy networks, the framework reveals architecture dependent differences in total transport scale and coherent displacement trace, while coherent displacement fraction is metric and resolution dependent.

Cross-subject EEG stress detection remains challenging because discriminative stress-related patterns are both subject-dependent and frequency-specific. Conventional Riemannian methods model spatial covariance mainly in the time domain, overlooking neural oscillations that are critical for high-level cognitive state decoding, while standard temporal tokenization often fragments inter-slice temporal coherence. To address these limitations, we propose \method{}, an Intra-Inter Riemannian Manifold Attention Network for EEG-based stress detection. \method{} constructs spatial covariance matrices independently at each frequency point and maps them to the SPD tangent space, preserving channel-wise geometry together with frequency-specific discriminative cues. It further introduces frequency cluster aggregation to select informative spectral components and reduce redundancy by forming compact, data-driven frequency clusters aligned with EEG rhythms. Finally, an intra-inter slice attention module adaptively integrates local slice-level spectral dynamics and global temporal context across EEG sequences. Experiments on three datasets show that \method{} consistently outperforms five state-of-the-art baselines, achieving up to 82.78% balanced accuracy while remaining efficient with only 1.60M parameters and 31.95M FLOPs.

Multi-animal tracking (MAT) is critical for wildlife monitoring and behavioral analysis, yet remains challenging due to uniform appearance, high density, and irregular motion. Existing methods typically follow heuristic- or query-based paradigms: the former relies on handcrafted geometric associations without end-to-end optimization, whereas the latter enables joint optimization but relies heavily on appearance embeddings. In such conditions, continuous geometric embeddings can be unstable, as small coordinate perturbations may disproportionately alter cross-frame attention weights, degrading identity association performance. To address this limitation, we propose HieDG, a Hierarchical Discrete Geometry-guided tracking framework that reformulates geometric dynamics as structured discrete representations within a query-based tracker. Instead of directly using raw geometric signals, HieDG employs a two-stage residual codebook to discretize position, scale, and velocity cues, transforming unstable continuous geometry into structured, stable discrete tokens. These tokens are aligned with visual embeddings and integrated into the tracking queries to enhance identity consistency. Extensive experiments on animal-specific benchmarks (AnimalTrack, BFT, and BuckTales) demonstrate state-of-the-art association performance with significant improvements in HOTA, AssA, and IDF1. Additional evaluations on generic multi-object tracking benchmarks, including DanceTrack and SportsMOT, show competitive performance, indicating the broader applicability of discretized geometric modeling beyond animal-specific scenarios.

Accurate modeling of dental crown morphology is fundamental for diagnosis, orthodontic planning, and computer-aided restoration design. However, datasets suitable for training such models are typically limited in size. We present ToothForge, a deep spectral generative framework that models dental crown geometries from compact, intrinsic representations. By operating in the spectral domain, ToothForge learns a latent manifold of 3D tooth shapes through synchronized spectral embeddings, ensuring consistent modeling across samples with varying connectivity. Spectral synchronization mitigates the instability of Laplace-Beltrami eigenbases and enables efficient learning in a low-dimensional space. The framework is thoroughly evaluated through robustness analysis, ablation studies, and benchmarking against PCA-based statistical shape models and point-based generative frameworks. Results show that synchronized spectral modeling achieves reconstruction and generative performance comparable to or exceeding spatial approaches, while maintaining compactness and geometric interpretability. Together, the compact synchronized coefficients and low-dimensional learning space make the framework particularly suitable for limited datasets, as often encountered in dental and medical domains, and applicable in real-world scenarios where guaranteeing consistent connectivity across shapes from various clinics is unrealistic.

LiDAR simultaneous localization and mapping (SLAM) relies on local geometric quantities such as covariances, correspondences, and surface structures. However, most existing pipelines rely on hand-crafted estimates of local geometry and use them as fixed inputs to LiDAR SLAM, which can make the estimated local geometry noisy and unstable in sparse regions of a point cloud or when using low-resolution LiDAR. To address this issue, this paper introduces a self-supervised framework that learns an explicit symbolic representation of local geometry and uses it to improve LiDAR SLAM recursively. Specifically, each point is represented as a Gaussian distribution, allowing local geometry to be described by a covariance. Without dense geometry labels or ground-truth poses, the framework learns by maximizing the likelihood of local geometry, with self-supervision derived from consistency relations over symbolic geometric representations, including predicted covariances, correspondences, and trajectory from SLAM. The learned geometry is then fed back into LiDAR SLAM, forming a reciprocal loop in which improved geometry enhances localization and mapping, and improved localization provides cleaner supervision for subsequent geometry reasoning. This framework is backend-agnostic and can be plugged into existing LiDAR SLAM pipelines without architectural changes. Experiments on KITTI under varying LiDAR resolutions show that the proposed method improves both odometry and global registration.

Classical universal approximation theorems (UAT) establish the expressive power of sigmoidal multilayer perceptrons, but they do not specify how the weights should be initialized. We study a supervised, data-dependent, geometry-aware initialization for one-hidden-layer sigmoidal MLPs that compiles labeled class geometry into network weights. The construction starts from the idea that sigmoid units can act as smooth half-space gates. For each class, we center the training samples at their mean, apply SVD to estimate principal directions and spectral scales, select retained directions by an energy threshold, and represent each retained direction by a pair of sigmoid slab gates. These class-specific gates are then concatenated into a shared hidden layer initialized directly from the training set. We also formulate a SVD-Mahalanobis 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 CIFAR-10 show that the proposed initializer produces a substantially more informative zero-epoch state than a matched task-agnostic Xavier reference, while full training reaches comparable final accuracy. Frozen-hidden experiments and neutral-head ablations further show that the class-wise SVD gates remain useful fixed features, even after the initially aligned output layer is replaced by a Xavier-initialized one.

Generalizing robotic manipulation to unseen objects remains challenging, as learning-based approaches require many demonstrations and fail in few-shot settings. Prior work transfers affordances through semantic retrieval, but semantics alone neglect geometric similarity, which is critical for manipulation. We propose GRAFT, a geometry-aware correspondence framework for zero-shot manipulation transfer using only one demonstration per object. Objects are represented as part-based graphs, where part-level descriptors support global instance retrieval and part correspondence, and vertex-level descriptors enable fine-grained contact point matching. For an unseen object, our method first retrieves the most functionally and geometrically similar instance from the demonstration buffer with aligned functional parts, and finally propagates the contact points through point-wise correspondence.

Grounding deictic gestures in natural images is fundamental to AR and human-robot collaboration, providing a basis for seamless spatial interaction. While Transformer-based visual models have achieved significant progress in general object detection, their global attention mechanisms often neglect micro-geometric relationships, degrading orientation accuracy. In pointing tasks, this deficiency manifests as an inability to accurately capture the pointing ray implied by finger poses, which results in pointing drift and localization ambiguity when dealing with distant or densely packed objects. To address this, we propose VistaRef, a framework designed to explicitly enhance spatial orientation awareness. First, we develop the Local Hand Entity Modeling (LHEM) module, which incorporates hand-pose embeddings to strengthen the model's capability to capture subtle finger deviations. Second, drawing inspiration from multi-view geometry, we construct the Geometric Ray Modeling (GRM) module to transform implicit orientation information into explicit spatial geometric features, guiding feature aggregation and deep fusion via attention mechanisms. Furthermore, we introduce a novel Orientation-Consistent Alignment Loss (OCAL) to synergistically supervise hand presence and pointing consistency, ensuring that all architectural improvements collectively serve the core objective of spatial localization. Experimental results demonstrate that VistaRef significantly outperforms the baseline, achieving a 14-point absolute gain in grounding accuracy. Qualitative analysis further confirms that VistaRef effectively models the geometric correlation from hand to target, bridging the spatial perception gap inherent in traditional Transformers for complex scenarios. Code: https://github.com/lingli1724/VistaRef.

Vision Foundation Models (VFMs) have significantly advanced dense feature matching, yet severe in-plane rotation remains a critical challenge. Existing solutions face a fundamental dilemma: data-driven methods require inefficient parameter scaling to implicitly learn rotations, whereas strictly equivariant networks lack the semantic capacity of modern VFMs. Consequently, current frameworks typically freeze VFMs and shift the entire burden of rotation generalization to the downstream decoder. To break this architectural bottleneck, we propose REDI-Match, an efficient framework driven by a novel Rotation-Equivariant Distillation (REDI) paradigm. Instead of relying on rotation data augmentation to establish rotational correspondences, REDI distills the non-equivariant semantic representations of a VFM into a lightweight, strictly rotation-equivariant encoder, leveraging an equivariant geometric architecture to constrain robust high-dimensional semantics. To fully exploit these features, we equip the decoder with an entropy-driven spatial alignment module. By evaluating discrete rotation hypotheses, this mechanism explicitly locks onto the canonical coordinate system, eliminating global ambiguity before continuous refinement. Extensive experiments demonstrate that REDI-Match establishes a new state-of-the-art (SOTA) across multiple benchmarks. Notably, it achieves a 13.89% absolute pose accuracy improvement on the highly challenging SatAst dataset while operating 1.9x faster than the current SOTA (RoMa v2), enabling real-time inference (~41 FPS) on a single RTX 4090 GPU. Code: https://github.com/YinjiGe/REDI-Match.

Humans localize places by integrating perceptual cues from vision with semantic reasoning from language, forming a scene understanding that is both intuitive and structured. Although existing geo-localization models have made substantial progress in cross-view and cross-modal settings, they are largely built upon point-to-point alignment, which is insufficient for joint vision-language queries. In such queries, visual and textual cues do not simply act as independent references, but jointly define a semantic subspace for locating the target. In this paper, we formulate vision-language geo-localization (VLGL) with joint image-text queries as a multi-anchor geometric alignment problem and propose a unified framework for this setting. To realize this formulation, we propose Multi-Anchor Projection Similarity (MAPS), a new metric which constructs an anchor plane from visual and textual query features in a high-dimensional space and measures similarity by the projection length of the target feature onto this plane. Unlike cosine similarity which evaluates isolated pairwise relations, MAPS captures the geometric consistency between the target feature and the joint query subspace, providing a more discriminative ranking criterion during retrieval. To make the learned representation consistent with this geometry, we further introduce a MAPS-based contrastive loss that drives target features toward the corresponding anchor plane. The proposed framework, similarity metric, and training objective jointly yield state-of-the-art performance in VLGL.

Traditional approaches place intelligence in the agent, whether as a learned policy or a search procedure. We instead place intelligence in the space itself: a scene induces a Riemannian metric on the configuration manifold, and action reduces to following the geodesics of that metric rather than invoking a separate planner or collision checker. A single Encoder-Router network realizes this idea through three complementary parameter groups -- frame parameters that orient the generators, modulation parameters that govern their spatial propagation, and basic coefficients that determine their strength. These groups combine through a shared semigroup-superposition mechanism to produce a single Riemannian metric field, yielding a compact architecture whose geometry scales naturally with scene complexity. Trained on a single two-obstacle scene, the model demonstrates robust zero-shot generalization across unseen obstacle configurations, with orders-of-magnitude separation between collision-free and obstacle-penetrating path costs.

Learning unsupervised representations of medical imaging cohorts can reveal anatomically meaningful prototypes without expert labels, which are often noisy and fail to capture true pathological heterogeneity. However, existing deep latent-variable models estimate Gaussian mixture priors via Euclidean averaging, producing prototypes that drift off the curved data manifold and degenerate as the number of sub-populations grows. We propose a manifold-anchored variational framework built on a geometry-aware Expectation-Maximization (EM) algorithm, whose M-step selects each sub-population prototype as the graph medoid with the highest diffusion centrality on a heat-kernel-weighted latent graph, ensuring that every prototype remains on-manifold. A Dirichlet energy regularizer enforces geometric smoothness of the latent space, and a per-sub-population uncertainty score enables label-free quality assessment. The manifold-anchored EM is a general-purpose geometric tool that extends standard EM and applies readily to other latent-variable models beyond this setting. On cardiac scar and brain MRI benchmarks, our framework attains the highest accuracy among all compared methods, produces the sharpest prototypes reported to date, and remains stable at large sub-population counts where all baselines degenerate. Code and implementation details are available at https://github.com/jr-xing/On-Manifold-Variational-Learning-with-Heat-Kernel-Priors.

We develop a framework for the information discarded by machine learning models whose inputs carry a Lie group action. Given a representation $π$ of a Lie group $G$ on a space $V$ and a learned function $f\colon V \to \mathbb{R}$, we define two objects measuring the symmetry invisible to $f$. The null fiber at a point $x \in V$ is the set $N_G(f,x) = \{g \in G : f(π(g^{-1}) \cdot x) = f(x)\}$ of group elements whose inverse action on $x$ is undetectable by $f$. When $N_G(f,x)$ is independent of $x$, it coincides with the stabilizer $\mathrm{Stab}_G(f)$, the largest subgroup of $G$ under which $f$ is invariant. For smooth maps to $\mathbb{R}$, the preimage theorem guarantees that null fibers have dimension at least $\dim G - 1$ at generic inputs, regardless of architecture. For compact groups acting on themselves, the Peter--Weyl theorem yields a spectral characterization of both objects in terms of the Fourier coefficient matrices of $f$. We show that null fiber elements can be computed efficiently via Newton iteration on the orbit map, at a cost comparable to a few gradient evaluations. Applications to data masking, model fingerprinting, and privacy-preserving computation are developed and tested experimentally on molecular property prediction under $\mathrm{SO}(3)$ and spherical image classification under the Möbius group $\mathrm{PSL}(2, \mathbb{C})$. The framework applies uniformly to classical neural networks and variational quantum circuits.

Computing geometric representations of data is a cornerstone of modern machine learning, typically achieved by training dual encoders which map queries and documents into a shared embedding space. Recent work of You et al. [NeurIPS '25] has extended this approach to hierarchical retrieval, where relevance is determined by the ancestor-descendant relationships in a Directed Acyclic Graph (DAG). While previous work has shown that valid embeddings exist when the number of descendants is small, these bounds degrade significantly for deep hierarchies, requiring dimensions as large as the total number of nodes. In this paper, we investigate compact reachability embeddings for more general graph classes and provide theoretical guarantees for representing hierarchies using embeddings whose dimension depends on structural graph parameters. We prove that for any directed tree, there exists a reachability embedding in constant dimension 3, independent of the tree's size or depth. We generalize this result to graphs characterized by treewidth $t$, constructing embeddings of dimension $O(t \log n)$, where $n$ is the number of nodes. Complementing these upper bounds, we provide matching or near-matching lower bounds, showing that dimension $Ω(n)$ is necessary for general DAGs and $Ω(t/\log(n/t))$ is required for graphs of treewidth $t$. We also obtain upper and lower bounds parameterized by the number of cross-edges in the DAG. We additionally show that our embeddings can be constructed on real world datasets, and that they give much smaller dimensions in high recall regimes compared to prior embeddings with theoretical guarantees.