Geometric Representation Learning

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28 papers in the last four weeks, up 155% on the four weeks before. 0.3% of all new papers.

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Latest papers 188

Oct 7, 2026cs.CV

$Δ$ Representation: Geometry Supervised Representation Learning of Phenotypes via Counterfactual Reasoning for Medical VLMs

Medical vision-language models (VLMs) have shown increasing potential for radiological image interpretation. Medical VLMs encode radiological images into visual representations that capture both anatomical and phenotypic information for diagnosis. Existing approaches improve pathological phenotype representations through semantic-guided representation alignment. However, pathological phenotypes arise as lesion-specific visual changes superimposed on underlying normal anatomy. Such semantic alignment approaches fail to model the phenotype-specific increment relative to the corresponding normal anatomical representation. To address this gap, we propose \textbf{

Δ

Representation}, a visual phenotype representation learning framework based on counterfactual reasoning for medical VLMs. It comprises \textbf{BaseAnatomy}, a geometry-supervised representation learning module, and \textbf{

Δ

Phenotype}, a counterfactual incremental representation learning module. BaseAnatomy provides fine-grained geometric supervision through spatial relationships across and within anatomical structures.

Δ

Phenotype computes the representation increment between lesion representations and their corresponding normal anatomical representations, and supervises increments associated with the same phenotype to cluster in the representation space. Experiments on \textit{ReXGroundingCT} and \textit{LIDC-IDRI} demonstrate that

Δ

Representation effectively structures pathological phenotype representations and improves lesion grounding and phenotype characterization accuracy in medical VLMs. Code is available at https://anonymous.4open.science/r/deltarep-CF6D.

Hao Wang, Qiwei Zeng, Jinghao Lin +6

The University of Sydney · Shanghai Jiao Tong University · Northeastern University +1

Medical Image Representation Learning Medical VLMs Medical Image Grounding Geometric Representation Learning

Oct 5, 2026cs.LG

Constant-Curvature Sliced Gromov-Wasserstein for Heterogeneous Cross-Curvature Alignment

Recent advances in representation learning have highlighted the utility of constant-curvature models, such as hyperbolic and spherical spaces, for modeling complex data. Mixed-curvature models further enhance this by integrating multiple constant-curvature components. However, these models typically learn each component space independently because spaces with different curvatures are inherently heterogeneous and lack a unified metric. Consequently, they lack explicit mechanisms to enforce geometric consistency across various spaces. Moreover, the problem of comparing probability distributions across mixed-curvature spaces remains unexplored. To compare distributions on heterogeneous spaces, Gromov-Wasserstein (GW) distances provide a principled framework by aligning their intra-space geometries. Building on this, we propose constant-curvature sliced Gromov-Wasserstein (CCSGW), a novel divergence for aligning distributions supported on heterogeneous constant-curvature spaces. We first introduce the missing geodesic-based one-dimensional projections for spherical spaces, and then extend sliced GW to constant-curvature spaces, enabling efficient and principled comparison across manifolds with different curvatures. This formulation preserves intrinsic geometric relationships while avoiding the high computational cost. We provide theoretical analysis showing that CCSGW controls intrinsic geometric discrepancy across heterogeneous spaces, promoting distribution-level geometric consistency. By integrating CCSGW into existing mixed-curvature learning tasks, including graph anomaly detection, graph node classification, and multimodal learning, we observe consistent performance gains across diverse settings.

Shanglin Li, Wenjing Lu, Muyang Li +2

BIFOLD, Germany · Technical University of Berlin, Germany · RIKEN AIP, Japan +4

Distribution Matching Geometric Representation Learning Optimal Transport

Oct 5, 2026cs.CV

Less Context, Better Geometry: Masked Geometric Encoder for Robust 3D Foundation Models

Recent progress in 3D foundation models has enabled rapid 3D reconstruction and camera calibration by leveraging learned 3D priors from vast amount of spatial data. However, the all-to-all global attention design leads to quadratic complexity and limits long-sequence inference; unconstrained cross-view interactions also can propagate unreliable evidence from occluded or visually similar but geometrically distant views. In this paper, We introduce a Masked Geometric Encoder (MGE), which promotes the learning of robust geometric representations under incomplete cross-view context. During training, MGE strategically drops frame tokens from global attention and distills from a pretrained full-context teacher model. This allows the model to learn an intrinsically richer per-frame representation while providing sufficient intermediate supervision to avoid performance degradation. Through extensive experiments, we show that MGE leads to much stronger performance under occlusion and doppelganger views while retaining high performance on standard benchmarks. Such a richer frame representation also leads to more effective token reduction during inference. To this end, we develop a novel Anchor-Guided Adaptive token merging technique that preserves representative anchor frames while jointly merging redundant tokens from the remaining views. Compared to other efficient inference approaches, we can achieve inference speedup while consistently maintaining higher reconstruction quality, particularly in limited-view settings.

Zhimin Shao, Xijun Liu, Zhaoliang Zhang +4

Department of Electrical and Computer Engineering Johns Hopkins University · Department of Data Science University of Virginia

3D Foundation Models Token Merging Multi-View 3D Reconstruction Sparse-View 3D Reconstruction Geometric Representation Learning

Oct 5, 2026cs.LG

On the Geometry of Multimodal Saturation: Riemannian VICReg

In self-supervised learning, a third modality should improve, or at least preserve, performance. Across nine image-text-tabular datasets, we show that it instead harms performance: the trimodal model underperforms its own best bimodal subset in 55.6% of paired runs under VICReg. The same failure occurs in 51.1% of paired runs under SimSiam. We call this failure multimodal saturation. We propose that the failure lies in the alignment geometry. Riemannian VICReg (R-VICReg) generalizes classical VICReg: it aligns views by squared geodesic distance on learnable negative-curvature product factors and recovers VICReg exactly as curvature vanishes. Over the same 45 paired runs, R-VICReg raises the probability that the third modality helps from 44.4% to 64.4%, with gains concentrated where VICReg saturates.

Nessim Ben Abbes, Duc Han Le, Sabri Mtibaa +1

Protectline · Télécom Paris

Self-Supervised Learning Geometric Representation Learning Hyperbolic Representation Learning Multimodal Learning

Oct 1, 2026cs.CV

GeoLatent: Geometry-Guided Latent Structuring with Routed Optimization for 3D Reasoning

Despite progress in vision-language models, 3D spatial reasoning from 2D images remains challenging. Text-based methods describe intermediate geometry with discrete tokens, limiting fidelity for continuous spatial relations. Continuous latents offer richer representations, but a single latent type does not explicitly separate the cues needed across spatial tasks. Decomposed spatial latents address this by representing position, direction, and global geometry separately under geometric supervision. Yet the geometry representation can still collapse toward one dominant direction, and unrestricted attention can leave the latents underused during answer learning. We introduce GeoLatent, combining Common--Residual Geometry Alignment (CR-GEO) with routed optimization to structure the geometry states while promoting latent-mediated answer learning. CR-GEO separates shared from residual teacher geometry; routed optimization jointly trains geometry and language, temporarily directs visual answer learning through the latents, and restores full attention with geometry supervision. In controlled comparisons, CR-GEO raises geometry effective rank from 1.00 to 3.87, while blocking latent readout at the bottleneck lowers direction accuracy from 89.1% to 25.8% on 128 fixed questions. After recovery, the differentiated geometry representation and latent-mediated visual route remain available alongside direct image access. GeoLatent achieves 73.0% on SPAR-Bench and 72.1% on SPBench, outperforming previously reported methods on both.

Yakun Zhu, Yi Bin, Yujuan Ding +5

Tongji University · The Hong Kong Polytechnic University

3D Spatial Reasoning Geometric Representation Learning Latent Visual Reasoning 3D VLMs

Oct 1, 2026cs.RO

Robot Learning on Discrete Surfaces: Theory and Applications

All the objects composing our world are enclosed within surfaces. Yet, most robot learning and motion generation frameworks treat surfaces as constraints ignoring their intrinsic geometry. This gap is acute for polyhedral meshes--the standard output of CAD and 3D reconstruction--whose discrete geometric structure remains unexploited. In this paper, we propose a unified discrete Riemannian framework that enables robot learning directly on polyhedral surface meshes. Using discrete differential geometry, we define logarithmic and exponential maps, parallel transport, and ambient-space projections that remain well-defined across faces, edges, and vertices. We instantiate the framework in three learning paradigms: (i) Dynamic Movement Primitives (DMPs), an improved exponential-map computation and a fixed-tangent-cone forcing-term encoding with parallel transport yield better cross-surface generalisation and stability over prior mesh-based approaches. (ii) Gaussian Process (GP), a geodesic-based kernel with practical admissibility control, enables regression at arbitrary mesh locations without smoothness assumptions. (iii) Riemannian Flow Matching (RFM), mesh-native operators improve generative quality over spectral baselines while reducing training time. The framework is validated in simulation against state-of-the-art methods and demonstrated on two real-robot scenarios: generalising user-drawn trajectories across different surfaces and planning polishing motions on RGB-D-reconstructed surfaces.

Matteo Dalle Vedove, Fares J. Abu-Dakka, Luigi Palopoli +2

Department of Industrial Engineering, Universit`a di Trento, Trento, Italy. · DRIM, Ph.D. of national interest in Robotics and Intelligent Machines. · Mechanical Engineering Program, Division of Engineering, New York University Abu Dhabi, Abu Dhabi, United Arab Emirates. +1

Geometric Representation Learning Robot Motion Generation Trajectory Representation Learning Robot Learning

Oct 1, 2026cs.CV

SmoothOperator: Enhancing Representations for Fine-grained Open-set Recognition via Modulated Label Smoothing

Open Set Recognition (OSR) aims to enable models to accurately classify known classes while rejecting samples from unseen classes. A key challenge in OSR lies in the inability to model the unbounded distribution of unknown classes during training, often leading to the misclassification of samples from these classes. Rather than modeling unknowns, recent work shapes the feature space so that known classes are compact and well separated, and spherical representation learning methods have achieved strong results this way. Label smoothing has been identified as one of the key drivers of this success, yet it applies the same coefficient to every training sample, regardless of how well each sample is already embedded. We show that the spherical representation learning objectives used in OSR share a single alignment--uniformity structure in which labels enter only through the alignment term. Label smoothing therefore acts as an alignment dial, and a fixed coefficient sets this dial to the same value for every sample. We propose a plug-in, SmoothOperator (SmoothOP), which sets the smoothing coefficient of each sample from its \textbf{prominence}, an embedding-space signal measuring how clearly the sample's own class stands out against its strongest competing class. Our method integrates into four existing spherical representation learning methods at minimal training overhead. SmoothOP assigns strong smoothing to samples with high prominence, which reduces their alignment and relaxes their pull. On the Semantic Shift Benchmark, SmoothOP-augmented variants generally outperform their base objectives across datasets, degrees of semantic shift, and OSR post-processors, with gains of up to 4.7% in AUROC, OSCR, and closed-set accuracy.

Thiru Thillai Nadarasar Bahavan, Yu Xia, Sachith Seneviratne +1

The University of Melbourne Parkville, VIC, Australia

Soft-Label Learning Open-Set Recognition Geometric Representation Learning

Sep 30, 2026cs.LG

A library for differentiable signal processing and machine learning on the sphere

The two-dimensional sphere embedded in three-dimensional Euclidean space S2, plays a central role in a variety of scientific and engineering domains, including geophysics, planetary science, geodesy, atmospheric physics, quantum chemistry, cosmology, and virtual reality, among many others. As machine learning increasingly permeates these fields, the demand grows for robust tools that process and model functions on the sphere, while respecting the inherent topological and symmetry properties of the domain. We present torch-harmonics, a comprehensive library that offers efficient, differentiable implementations of advanced signal processing and machine learning (ML) methods for spherical data. These include the spherical harmonic transform (SHT), the spherical analogue of the Fourier transform, vector spherical harmonics, discrete-continuous and spectral convolutions, as well as both global and neighborhood spherical attention mechanisms. Beyond traditional representations, torch-harmonics provides the building blocks for state-of-the-art spherical ML architectures such as spherical transformers in order to enable scalable, rotationally-aware learning and inference in modern scientific and engineering applications.

Thorsten Kurth, Max Rietmann, Mauro Bisson +6

NVIDIA Corporation · California Institute of Technology

Geometric Representation Learning Spherical Harmonics Scientific ML

Sep 30, 2026cs.CV

How to Reduce Localization Ambiguity? Geometry-Semantic Constrained BEV Representation Learning for Satellite-Ground Localization

Satellite-ground localization estimates the planar position and yaw orientation of a ground camera within a geo-referenced satellite image. Most recent methods map ground and satellite features into a shared bird's-eye-view (BEV) space and establish spatial correspondences. However, insufficient depth constraints can assign one ground feature to different distances along a viewing direction, creating geometric ambiguity in BEV feature placement. Similar appearances at different locations can also create descriptor matching ambiguity, while existing descriptor learning lacks explicit semantic supervision to distinguish them. We propose GeoSem-BEV, a geometry-semantic constrained BEV representation learning method. Radial depth supervision constrains distance assignment, and vertical height supervision constrains height aggregation. Shared explicit semantic supervision promotes consistent semantic predictions across views and helps distinguish locations with similar semantics. These constraints improve feature placement and descriptor discriminability, enhancing state-of-the-art BEV localization models. On VIGOR with unknown orientation, GeoSem-BEV reduces mean orientation error by 37.2% and 38.1% in the cross-area and same-area settings, respectively. The corresponding errors are reduced by 10.8% and 15.6% on DReSS-D. On KITTI-CVL, it reduces same-area mean orientation error by 26.8% under 10 degree orientation noise.

Junming Feng, Panwang Xia, Qiong Wu +8

The Hong Kong Polytechnic University, Hong Kong · Southern University of Science and Technology · Wuhan University, Wuhan, China +2

Representation Learning Cross-View Geo-Localization Geospatial Representation Learning Geometric Representation Learning Bird's-Eye-View Representation

Sep 30, 2026cs.DS

Optimal VC Dimension of Contrastive Learning with Margin

Contrastive learning is a successful paradigm for learning

d

-dimensional geometric representations from a collection of anchor--positive--negative'' triplets $(i,j^{+},k^{-})$, indicating that item

i

is closer to

j

than to

k

.'' Despite its success, understanding why contrastive learning leads to representations of high \textit{generalization} quality---beyond the often pessimistic predictions from PAC-learning---remains a central question. Recently, \citet*{alon2024optimal} proved that, for PAC-learning

d

-dimensional Euclidean representations of

n

-point datasets,

Θ(\min(nd, n^2))

triplets are necessary and sufficient, while they posed as an open question whether their VC dimension bounds for the more realistic setting of \textit{contrastive learning with a margin} can be improved. For a margin parameter

α>0

, a triplet

(i,j^{+},k^{-})_α

is satisfied by the embedding

φ:[n]\rightarrow \mathbb{R}^{d}

, if

\|φ(i)-φ(k)\|_2>(1+α)\cdot\|φ(i)-φ(j)\|_2

. In this work, we resolve their question by proving that the VC dimension of contrastive learning under any margin

α\in(0,1)

is in fact

O(n/α^2)

, improving on the previous bound of

O(n\log(n)/α^2)

. We also establish that the bounds are optimal up to constant factors, by providing a matching lower bound of

Ω(\frac{n}{α^2})

(the previously known lower bound was

Ω(\frac{n}α)

), for

α\geq \max(n^{-1/2},d^{-1/2})

.

Dionysis Arvanitakis, Vaggos Chatziafratis, Yiyuan Luo +1

Northwestern University · UC Santa Cruz

Contrastive Learning Geometric Representation Learning

Sep 29, 2026cs.CV

Minkowski Attractor Networks: Closed-Form Hyperbolic Flows for Visual Representations

Geometric representation learning predominantly scaffolds representations onto flat Euclidean subspaces or compact product tori (

\mathbb{T}^K

). However, flat manifolds possess vanishing curvature and polynomial volume growth, inherently suffering from metric distortion when embedding multi-scale, tree-like visual hierarchies. While hyperbolic spaces (

\mathbb{H}^m

) circumvent this via constant negative curvature (

K<0

) and exponential volume expansion, prior hyperbolic deep architectures are hindered by computationally cumbersome Riemannian optimization, non-linear gyrovector calculus, and floating-point instabilities. In this work, we introduce \textbf{Minkowski Attractor Networks (MAN)}, an operator-splitting-inspired framework that embeds representations within pseudo-Riemannian Minkowski spacetime (

\mathbb{R}^{1,m}

). By framing hyperbolic manifolds as quadric level sets, MAN resolves hyperbolic geometry by combining linear Lorentz group transport with non-linear cone lifting and closed-form radial rescaling, evaluating in a single forward pass without numerical ODE solvers or iterative retractions. We establish \textbf{MAN-2D} (

\mathbb{R}^{1,1} \to \mathbb{H}^1

) as our primary, high-throughput visual backbone, which maximizes channel factorization granularity into

D/2

independent two-dimensional Minkowski blocks. We further formulate \textbf{MAN-4D} (

\mathbb{R}^{1,3} \to \mathbb{H}^3

) as a spacetime extension, leveraging a commuting Cartan-subalgebra parameterization of

\mathrm{SO}^+(1,3)

to evaluate 4D Lorentz isometries via two commuting 2D planar maps without matrix-exponential overhead.

Zhongping Ji

Visual Representation Learning Hierarchical Representation Learning Geometric Representation Learning Hyperbolic Representation Learning

Sep 28, 2026cs.RO

ATLAS: Aligned Transport of Latent Structure for Reliable World Model Planning

Latent world models rely on representation geometry for planning, yet regularizing the latent marginal alone does not determine the state-to-state relationships used for action selection. We show that this can cause planning-relevant novelty structure to be weakened as representations are transformed into the final latent used by the planner. We introduce Aligned Transport of Latent Structure (ATLAS), a training objective that explicitly preserves relational geometry while calibrating the global latent distribution. ATLAS transfers normalized pairwise structure from an informative encoder representation to the planning latent and uses Wasserstein embedding matching (WEMReg) to calibrate its marginal through one-dimensional Wasserstein-2 transport. Our analysis shows that relational preservation and marginal calibration impose non-redundant constraints, and connects finite-candidate planning stability to relational distortion, latent-scale mismatch, and prediction error. Instantiated in LeWM, ATLAS improves mean goal-reaching success across PushT, TwoRoom, and OGBench-Cube on both lower- and higher-novelty evaluation subsets, with the largest gain on higher-novelty TwoRoom episodes. Representation and rollout diagnostics further show stronger novelty-related structure in the planning latent, improved marginal calibration, and lower multi-step prediction error. Together, these results highlight preservation of planning-relevant latent geometry as an important ingredient for reliable world-model planning. Code is available at https://anonymous.4open.science/r/atlas-world-model-72C4/.

Ke Fang, Yupu Yao, Lu Cheng

Pennsylvania State University

World Models World Model-Based Planning Geometric Representation Learning Model-Based Planning Latent World Models

Sep 28, 2026cs.AI

Building Transformation Layers for Riemannian Neural Networks

Recently, deep neural networks on manifold-valued representations have garnered significant attention across various machine learning applications. One recent focus is the generalization of Euclidean fully connected (FC) and convolutional layers to non-Euclidean geometries. However, previous approaches typically focus on a few selected manifolds and rely on specific properties of the target manifold. In contrast, this work proposes a framework for constructing FC and convolutional layers over computationally tractable Riemannian spaces. This framework incorporates several previous FC layers across different geometries as special cases and is instantiated on ten representative manifolds, including three hyperbolic models, five geometries of the symmetric positive definite (SPD) manifold, and two Grassmannian perspectives. Experiments on different manifolds demonstrate the effectiveness and applicability of our approach. Code can be found at https://github.com/GitZH-Chen/RieTrans.

Ziheng Chen

University of Trento · MPI for Intelligent Systems, Tübingen

Geometric Representation Learning

Sep 28, 2026cs.AI

AbGaze: Attentive Geometric Representation Learning for End-to-End Antibody Design

Computational antibody design requires representations that capture the geometric patterns underlying antigen--antibody interactions, yet existing approaches often rely on scalar distances or surface-intrinsic features, leaving cross-molecular geometry largely implicit. We present AbGaze, an end-to-end antibody design framework based on attentive geometric representation learning, which encodes distance, spatial direction, and surface-normal orientation of antigen surfaces relative to antibody-residue local frames, and adaptively aggregates these geometric interactions according to their interfacial context. The learned interaction representation is shared across multi-CDR co-design, complex structure prediction, and affinity optimization, with local-frame geometric supervision further constraining the representation. AbGaze outperforms prior methods across all three tasks: relative to the second-best method, it improves amino-acid recovery by 7.1% and reduces structural error by 14.9% on average over the six CDRs, improves interface docking quality (DockQ) by 6.6%, and raises the affinity improvement rate (IMP) by 32.5%.

Jiashuo Wang, Siqi Fan, Yizhen Luo +1

Institute for AI Industry Research (AIR), Tsinghua University · Department of Computer Science and Technology, Tsinghua University · PharMolix Inc.

Geometric Representation Learning Molecular Generation Protein Design

Sep 28, 2026cs.LG

Learning Propagation Geometry from Message-Passing Feedback

Learning local geometry enables graph neural networks (GNNs) to adapt how they compare and integrate neighborhood information. However, estimating geometry from aggregated representations can overlook variation among individual messages and dependencies across feature dimensions. We propose GeoF, a recurrent framework that jointly evolves node features and propagation geometry through message-passing feedback. Each node maintains a local symmetric positive-definite geometry, initialized from a structure-aware prototype atlas and parameterized in block log-triangular coordinates. At each step, the geometry determines neighborhood weights, while triangular frame transport maps transformed source messages into the target node's local coordinates before aggregation. Weighted second-order statistics of residuals between aligned messages and the transformed target state capture directional variation and within-block dependencies, yielding a geometric update target. A shared controller learns complementary corrections through task supervision. A bounded log-triangular update combines these corrections, the target, and the previous geometric state while preserving positive definiteness. The geometry governs subsequent propagation, closing the feedback loop. With parameters shared across recurrent steps, task-specific readouts support node classification, link prediction, and graph classification. Experiments on benchmark datasets show that GeoF consistently outperforms state-of-the-art GNN baselines.

Yingxu Wang, Kunyu Zhang, Xinwang Liu +4

The Chinese University of Hong Kong · The Education University of Hong Kong · National University of Defense Technology +3

Graph Neural Networks Message Passing Neural Networks Geometric Representation Learning

Sep 27, 2026cs.CV

PGL-3D: Towards Progressive Geometric Learning for 3D Visual Query Localization

3D Visual Query Localization (3DVQL) retrieves the latest contiguous occurrence of a queried object in an RGB--point-cloud sequence and predicts a 9-DoF cuboid for every response frame. The query is captured independently of the search sequence, so its annotated pose may differ from how the object appears in the search frames. The benchmark baseline predicts cuboids after feature modeling, leaving their geometry unused for subsequent feature refinement. We investigate whether complete intermediate cuboids can improve query and proposal representations before final decoding. We introduce Progressive Geometric Learning for 3DVQL (PGL-3D), a predict--select--refine--re-predict framework that uses intermediate cuboids to guide the aggregation of search evidence and update query and proposal representations. A shared head first predicts a complete cuboid for every proposal. Query--Tube--Memory (QTM) then selects reference observations by combining proposal association, cuboid quality, frame response, and target absence, since association confidence alone establishes neither target presence nor geometric accuracy. The center, size, and orientation of each selected cuboid define soft pooling weights over query-conditioned proposal features. The pooled memory updates the query and proposal representations, and the head re-predicts from the updated features. A training-only objective, ST-D9O, supervises cuboid geometry at every stage by adding boundary, signed-distance, and soft-overlap terms to parameter regression. PGL-3D achieves a mean stAP of

0.270 \pm 0.004

on 3DVQL, compared with

0.044

reported for LaF. Ablations support the benefits of geometry-guided feature updates, while stage-wise analyses show improved cuboid accuracy. Replacing the geometry objective in our PROT3D reproduction with ST-D9O improves mAO on GSOT3D from

21.63\%

to

25.78\%

. Our code and models will be released.

Liang Peng, Shizhuo Mu, Bohan Tan +6

School of Computer Science, National Engineering Research Center for Multimedia Software, Institute of Artificial Intelligence, Hubei Key Laboratory of Multimedia and Network Communication Engineering, Wuhan University · University of North Texas · Institute of Software, Chinese Academy of Sciences

Geometric Representation Learning 3D Representation Learning 3D Visual Grounding

Sep 24, 2026cs.CV

Learning a Flow to Self-Supervised Representations

Explicit geometric references offer a direct way to structure self-supervised representations. Existing adversarial distribution-matching formulations, however, require costly encoder-critic optimization. We introduce Flow-Based Distribution Matching (FBDM), a non-adversarial framework that learns this reference-directed geometry through spherical conditional velocity regression. An ETF-inspired reference allows its number of components K' to exceed the auxiliary flow dimension d* while retaining structured geometric separation. We assign both augmented views of each image to the same target, while limiting how many images each reference center can receive. An explicit alignment loss further pulls the two views' representations closer together. Experiments across benchmarks ranging from CIFAR to ImageNet show that FBDM achieves performance nearly on par with DM and remains competitive with existing SSL methods. Matched training-cost comparisons show a 1.48- to 1.83-fold speedup over DM with a negligible increase in GPU memory usage. We also provide a theoretical explanation for the usefulness of the learned representations: under stated conditions, we bound the downstream misclassification rate in terms of the FBDM pretraining loss.

Yuling Jiao, Wensen Ma, Houduo Qi +1

School of Artificial Intelligence, National Center for Applied Mathematics in Hubei, Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, China. · Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong SAR, China. · Department of Data Science and Artificial Intelligence, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong SAR, China.

Flow Matching Representation Learning Geometric Representation Learning Self-Supervised Visual Representation Learning

Sep 23, 2026cs.CV

Depth-Guided Contrastive Learning for 2D Representations with 3D Spatial Awareness

Standard contrastive learning frameworks are mainly designed from a semantic perspective, yet learning 2D visual representations that preserve 3D spatial structure is also important for scene understanding. In this work, we propose Depth-Guided Contrastive Learning (DGCL), a simple auxiliary objective that injects 3D spatial awareness into 2D contrastive representation learning. Our key idea is to use depth to convert local 3D proximity into contrastive similarity: pixels that are closer in 3D space are encouraged to have more similar representations than pixels that are farther apart. Instead of relying on absolute depth values, DGCL formulates supervision through relative 3D distance comparisons among randomly sampled pixels, making the objective invariant to depth scale, efficient to compute, and easy to integrate into existing contrastive frameworks. Experiments across different datasets and models show that DGCL consistently improves 2D representation learning and benefits semantic downstream tasks by stronger spatial and geometric understanding. The code is available on https://github.com/LeungTsang/DGCL.

Liang Zeng, Maarten Vergauwen

KU Leuven Belgium

Contrastive Learning Visual Representation Learning Geometric Representation Learning

Sep 23, 2026cs.CV

Task-Induced Riemannian Metrics for Vision Transformer Feature Spaces

Methods operating on Vision Transformer (ViT) feature spaces typically rely on Euclidean distance or cosine similarity. This assumes that every direction is equally meaningful, but there is no reason to believe the true task geometry has this property. The task-sensitive geometry of the feature space is given by the pullback metric

g(F) = J(F)^\top J(F)

, where

J

is the Jacobian of the decoder's output fed to a task-specific distance, with respect to the features. Storing the full

g

is infeasible at modern scales, and for dense outputs such as depth maps even forming

J

is impractical. We show that whether a low-rank approximation of this metric can be learned depends on the model-decoder pair, and we characterize this with a matrix-free diagnostic

κ_{cap}(r)

computable with a low number of Jacobian-vector products. For tractable pairs, we develop the Spectral Pullback Network (SPN), which learns a low-rank version of the metric from randomized power iteration, and we distill it into a

310

K-parameter importance head that predicts token importance directly from the features. When the Jacobian spectrum is too spread out for a low-rank approximation, passing the decoder's input features through a VAE bottleneck can restore tractability. Across DPT, DINOv2, CLIP, and VGGT backbones,

κ_{cap}(r)

predicts which learned-metric architectures are viable. The importance head reaches Spearman

ρ= 0.998

on DINOv2 CLS, and our geometric token pruning reduces the additional depth error of ToMe-based token selection by

25\%

on DPT depth at prune ratio

0.5

, without fine-tuning the ViT. Project page: https://cyberiada.github.io/TaskInducedViTs/

Andrew Bond, Ege Erdem Özlü, Tuna Çimen +4

Imperial College London London, UK · Hacettepe University Ankara, Türkiye

Vision Transformer Geometric Representation Learning Metric Learning Efficient ViTs

Sep 23, 2026cs.CV

Geometry-Conditioned Visual Place Recognition in Natural Environments

Visual Place Recognition (VPR) in natural environments remains challenging due to repetitive vegetation, sparse distinctive landmarks, and substantial appearance and viewpoint variation across traversals. While visual observations of the same place can change considerably, their underlying spatial structure is often more persistent. We exploit this complementary geometric consistency through Depth-Aware Distillation (DAD), which conditions the token representations of a pretrained Vision Foundation Model (VFM) on geometry inferred by a Geometric Foundation Model (GFM), without any depth sensor. Rather than treating geometry as an additional input modality, DAD projects image-aligned depth into the VFM token space and selectively modulates visual representations through channel-wise geometric conditioning. A two-stage teacher-guided learning strategy first anchors the geometry-conditioned representation to the pretrained appearance space, before refining it for place discrimination. Evaluated on the WildCross benchmark, DAD improves average inter-sequence Recall@1 from 61.41% to 66.37% and Recall@5 from 65.86% to 72.49% over a matched appearance-only baseline, with the largest gains under reverse traversal and long-term appearance variation. These results show that GFM-derived geometry can provide a persistent structural prior for VPR when visual appearance becomes unreliable.

Walter Nedov, Saimunur Rahman, Kavindie Katuwandeniya +3

CSIRO Robotics, CSIRO, Australia · Queensland University of Technology, Australia

Visual Representation Learning Visual Place Recognition Vision Foundation Model Adaptation Geometric Representation Learning

Sep 22, 2026cs.AI

Canonical locks that encode part-whole hierarchies

One of the challenges in representational learning is how to encode part-whole hierarchies in a neural net. Prior works rely on flattening tree-like structures into string-like sequences and training a sequence-to-sequence model via autoregression. While such a representation works for parse-trees in NLP, it is not entirely clear how to make it work for images. Thus, we propose a geometric primitive called canonical locks. The key idea is that parts/wholes can be modelled as higher-dimensional vectors (

d \geq 4

), and information can be encoded in their relative phase differences. Inductively, the net consists of positionally-bound bottom-up and top-down neural fields, which drive each other to achieve a state of thermal equilibrium. Additionally, we show the existence of a few symmetrical configurations in the net. The computational iterations taken to break these symmetries depend on the angle between parts/wholes arranged on a disk (or more precisely a ring) in higher dimensions. It also appears to have connections to the psychological phenomenon of mental rotation.

Rajat Modi, Yogesh Singh Rawat

Department of Computer Science, University of Central Florida, United States.

Visual Representation Learning Hierarchical Representation Learning Geometric Representation Learning Compositional Representation Learning

Sep 21, 2026cs.CV

Ananke: Contractive Torus Attractor Networks

We introduce Ananke, a representation-learning framework that scaffolds latent representations onto a structured product-torus prior, and its flagship visual backbone realization, Contractive Torus Attractor Networks (CTAN). By factorizing high-dimensional latent spaces into an orthogonal direct sum of two-dimensional phase planes (

\bigoplus_{k=1}^K \R^2

), Ananke coordinates feature updates via a decoupled dual-phase continuous flow: skew-symmetric Hamiltonian transport moves features tangentially along energy level sets to preserve semantic phase invariants, while signed gradient dissipation contracts transverse perturbations normally toward target invariant manifolds. For circular potential families with frozen parameters, logarithmic radial feedback yields the Exact Log-Symplectic Flow (ELSF), an analytical closed-form mapping with exact exponential decay of log-radius error that evaluates in a single forward pass without numerical integration. We establish local input-to-state bounds for level-set deviations and log-radius errors, and characterize the normal hyperbolicity and persistence of the ideal product torus under bounded perturbations. We further formulate the architecture through Lie--Trotter operator splitting, unifying spatial depthwise diffusion with local manifold contraction, and analyze both exact trigonometric flows and hardware-friendly symplectic dual-shear variants. Across natural image benchmarks (CIFAR-100) and clinically challenging endoscopy datasets (Kvasir-v2), CTAN demonstrates exceptional parameter efficiency: an ultra-compact hierarchical model with merely 0.27M parameters achieves 90.52% accuracy on Kvasir-v2, outperforming 25M+ baselines (ResNet-50, DenseNet-161) by nearly two orders of magnitude in capacity, while scaled variants attain 80.32% top-1 accuracy on CIFAR-100.

Zhongping Ji

Visual Representation Learning Geometric Representation Learning

Sep 20, 2026cs.CV

GeoBalance: Geometry-Aware Monitoring and Reconstruction with Asymmetric Optimization for Balanced Multimodal Learning

Multimodal classifiers can converge to modality-dominant solutions in which one modality dominates the joint prediction, suppressing the learning of others. Existing balancing methods mainly adjust losses, gradients, or modality contributions, largely treating modality imbalance as an optimization problem while implicitly treating the weak modality as under-optimized but representationally intact. In this work, we find that this assumption does not always hold, as persistent modality dominance can induce a representation-level collapse of the weak modality, which we term \emph{manifold modality collapse} (MMC). MMC manifests as a coupled geometric degradation in which weak-modality representations collapse onto fewer directions within each class and become less separable across classes. Motivated by this observation, we propose \emph{GeoBalance}, a geometry-aware framework that monitors these two geometric properties and reconstructs the weak modality representation only when it exhibits signs of MMC. Once triggered, GeoBalance uses a fixed Simplex-ETF class scaffold and spectral regularization to restore class separation while preventing collapse onto a few feature directions. To preserve reconstruction during joint training, asymmetric gradient projection removes the joint-gradient component conflicting with reconstruction, leaving non-conflicting optimization unchanged. Extensive experiments across six multimodal benchmarks demonstrate great improvements over competitive balancing methods, validating its effectiveness.

Zechang Xiong, Da Li, Rong Yin +7

Beihang University, Beijing, China · Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China · Amap, Alibaba Group, Beijing, China +3

Modality Imbalance Geometric Representation Learning Multimodal Optimization Multimodal Classification

Sep 16, 2026cs.RO

InterMASH: A Unified Geometric Representation for Grasp Synthesis

Grasp synthesis aims to generate stable and physically plausible hand--object interactions, and has become a fundamental problem in both human hand modeling and robotic manipulation. However, a unified representation across human and robotic hands is still lacking, mainly due to differences in hand morphology and surface modeling. Prior methods typically rely on either contact maps or dense implicit descriptors to represent interaction, but these representations are often incomplete or computationally expensive and redundant. We propose InterMASH, a unified geometric representation that establishes cross-embodiment correspondence using sphere-fixed anchors. At each anchor, low-degree spherical harmonics compactly encode local hand geometry, object geometry, and contact, forming an explicit and interpretable token sequence. Building on this natively tokenized structure, we introduce a conditional Diffusion Transformer that operates directly in the proposed InterMASH representation space and jointly generates hand geometry and contact, improving consistency and physical plausibility. Our method achieves competitive performance with state-of-the-art methods on key physical feasibility metrics in a large-scale ShadowHand benchmark, supports joint training across multiple hands, and shows that cross-embodiment fine-tuning with human grasp data can improve robotic grasp success and diversity. Project page is available at https://inter-mash.github.io/.

Xuanze Yang, Yumeng Liu, Haiyang Xin +5

University of Science and Technology of China, Hefei, China · Jiangsu Key Laboratory of AI for Industries and Institute of AI for Industries Chinese Academy of Sciences, Nanjing, China · Shenzhen University, Shenzhen, China

HOI Generation Robotic Grasping Cross-Embodiment Robot Learning Geometric Representation Learning

Sep 16, 2026cs.RO

ReRadar: Robust Radar Global Localization via Rotation-Equivariant Descriptor Learning

Global localization with scanning millimeter-wave radar remains challenging because place-recognition descriptors often discard spatial structure needed for accurate pose retrieval. We present ReRadar, a radar global localization pipeline that extracts rotation-equivariant intermediate features using steerable convolutional neural networks, forms rotation-invariant descriptors through group pooling and NetVLAD aggregation, and combines descriptor retrieval with landmark-based matching to estimate the robot's three-degree-of-freedom (3-DoF) pose. Across fixed database-query evaluations, ReRadar with target-dataset adaptation achieves 99.37% Recall@1 on OORD Bellmouth, 91.44% Recall@1 with 80.99% F1_max on Mulran DCC01, and 99.38% Recall@1 on falling-snow Boreas sequence. Without target-dataset data, the cross-dataset model reaches 98.07% Recall@1 on OORD, performing comparably to the evaluated state-of-the-art methods.

Duc Manh Nguyen, Truong Giang Dao, Gia Nghiem Luong +2

School of Electrical and Electronic Engineering, Hanoi University of Science and Technology, Vietnam · School of Information and Communications Technology, Hanoi University of Science and Technology, Vietnam

Millimeter-Wave Radar mmWave Radar Sensing Geometric Representation Learning Robot Localization

Sep 15, 2026cs.CV

GeoLAM: Learning Geometry-Grounded Latent Actions from Unlabeled Human Videos

Human videos provide rich manipulation experience, but extracting action representations that preserve useful motion remains challenging. Visual reconstruction alone can entangle manipulation-related motion with appearance changes and camera movement. We present GeoLAM, a framework for learning geometry-grounded latent actions from action-free human videos. GeoLAM combines future-frame reconstruction through a frozen geometric feature hierarchy with motion supervision from a training-only 4D geometry teacher. The geometric representation provides a structural prior, while the teacher's predictions yield spatially pooled targets capturing 3D displacement, residual image-plane motion, and surface-orientation changes. Visibility and confidence weighting reduces the contribution of unreliable estimates, encouraging continuous latent actions to retain geometric motion without explicit hand-pose or hand-trajectory annotations. After video pretraining without action labels, the learned representation provides transition targets for a world-action model trained on action-labeled robot demonstrations. The model jointly denoises latent actions and executable action chunks, with future-video prediction used only as an auxiliary training task. Deployment therefore requires neither the geometry teacher nor future-video generation. Evaluations on a latent-action benchmark and robotic manipulation tasks demonstrate the strong performance of GeoLAM.

Yifan Xie, Hekun Tian, Jinkun Liu +3

Tsinghua University · Beijing Institute of Technology · Fudan University

Latent Action Learning Geometric Representation Learning Video Representation Learning Robotic Manipulation Motion Representation Learning

Sep 14, 2026cs.CV

SURE-Map: Self-Correcting Streaming Geometric Foundation Models

Streaming geometric foundation models are emerging as a compelling alternative to SLAM systems. Yet this streaming nature introduces a fundamental issue: each prediction is made from limited context, which is vulnerable to dynamic objects and weak textures. Small local errors accumulate into severe geometric distortion and long-horizon scale drift. We argue that reliable streaming reconstruction requires geometric foundation models to be not only predictive, but also self-correcting. We introduce SURE-Map, a self-correcting framework built upon two complementary principles. First, we explicitly model cross-view geometric uncertainty. Unlike conventional depth or point confidence, which primarily reflects the reliability of individual-view prediction, our uncertainty directly measures whether the jointly predicted pose and depth induce geometrically consistent cross-view pixel correspondences. Second, because local correction alone cannot eliminate slowly accumulating scale errors, we introduce multi-timescale self-correction: fast consecutive-frame inference preserves streaming efficiency, while sparse keyframe-window inference provides longer-range geometric evidence to periodically recalibrate the scale of recent trajectories. SURE-Map establishes new state-of-the-art performance for online feed-forward reconstruction across long-horizon benchmarks, reducing ATE-RMSE from 24.00 to 17.24 m on KITTI, 5.11 to 4.74 m on Oxford Spires, and 31.37 to 28.58 m on VBR, with further improvements to 15.17, 4.63, and 22.12 m when incorporating loop-closure refinement. Project page: https://mingkai-liu.github.io/projects/sure-map/.

Mingkai Liu, Hao Zhao, Xingxing Zuo

Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), UAE. · Peking University, China. · Tsinghua University, China.

Streaming 3D Reconstruction Uncertainty Quantification Geometric Representation Learning Feed-Forward 3D Reconstruction

Sep 14, 2026cs.LG

$\mathbb{SL}(n)$ Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition

Mixed-curvature representation learning seeks to capture rich geometric structures that cannot be adequately modeled by a single curvature regime. Existing approaches largely rely on product manifolds, which require manually specifying how different curvature spaces are combined and separate their curvature contributions across factors. We introduce the

\mathbb{SL}(n)

space, a representation geometry defined by the simple

\det(A)=1

constraint and a left invariant Schatten-

p

Finsler structure. Despite this minimal construction,

\mathbb{SL}(n)

exhibits pointwise negative, zero, and positive flag curvature around a common flagpole, while its mixed-curvature and curvature-coupling capacities are asymptotically maximal relative to the intrinsic geometric upper bound. Beyond geometry, its noncommutative group structure provides inherent order sensitivity, and its non-nilpotent Lie algebra admits nonzero nested Lie brackets at arbitrary depth, enabling deep order-aware composition. Empirically,

\mathbb{SL}(n)

consistently outperforms a broad range of representation manifold baselines across graph benchmarks at different scales. It reduces average distortion over the strongest baselines by

44.3\%

on KEGG and

40.5\%

on HumanCyc, and improves Hits@20 by

42.8\%

on OGBL-PPA. Experiments on Flickr30k-Order further support its ability to capture higher order dependencies from ordered composition. Together, these results show how a seemingly simple structural constraint can yield unexpectedly rich geometry, capacity, and composition within a unified representation space.

Xingrun Li, Yusuke Mukuta, Xin Yang +2

The University of Tokyo · Stanford University

Graph Representation Learning Geometric Representation Learning Lie Group Methods

Sep 14, 2026cs.LG

Geometric-to-Semantic Spherical Transfer Learning for Cortical Sulci Labeling

Deep learning on cortical surfaces faces a dilemma: capturing the complex topology of over 60 nomenclature-dependent sulci per hemisphere requires high-capacity models, yet the extreme scarcity of expert annotations (

N=62

subjects) inevitably causes overfitting. Standard supervised approaches fail to generalize in this data-scarce regime, particularly for variable and small sulci where topological ambiguity is high. To overcome this limitation, we introduce a Geometric-to-Semantic Spherical Transfer Learning framework. First, we leverage massive unlabeled data (UK Biobank,

\approx

30,000 subjects) to pre-train a spherical encoder using a locally-optimized strategy. By relying solely on continuous surface features (curvature and depth), the relevance of this pre-training is confirmed by the model's ability to detect localized and rare topological traits, such as sulcal interruptions. The downstream labeling task, however, introduces extracted sulcal fundi (lines) as an explicit semantic input. To bridge this dimensional domain gap (from purely geometric to semantic) without causing catastrophic forgetting, these anatomical lines are integrated into the pre-trained backbone via a soft-initialized Topological Prior Injector. Our experiments demonstrate that this approach outperforms fully supervised baselines trained from scratch, achieving a mean Dice of 0.77. Crucially, a local analysis reveals that the self-supervised geometric priors yield the largest performance gains on variable and tertiary sulci (up to 14.8%), confirming that learning the cortex shape is highly beneficial for identifying its rarest parts.

Saeb Tounsi, Joël Chavas, Pietro Gori +3

Paris-Saclay University, CEA, NeuroSpin, Baobab, Saclay, France · LTCI, Télécom Paris, Institut Polytechnique de Paris, France

Self-Supervised Pre-Training Geometric Representation Learning Cross-Domain Transfer Learning Neuroimaging Medical Image Representation Learning

Sep 9, 2026cs.LG

CoGe-GCD: Reframing Generalized Category Discovery with Compositional Generalization

Generalized Category Discovery (GCD) assigns unlabeled instances, mixed with labeled data, to known or novel categories, requiring human-like compositional reasoning: reusing primitives learned from known classes and deciding when new combinations imply new categories. Existing GCD methods operate on unstructured token features and struggle to extrapolate to novel compositions. We propose CoGe-GCD, which rethinks GCD through compositional generalization with two coupled stages. (i) Compositional Perception structures patch tokens by mapping them to a small vocabulary of primitives and refining token embeddings via competitive token-primitive assignment and information passing, yielding coherent groups for discovery. (ii) Generalizing Induction exploits the induced geometric structure and applies a structure-preserving calibration over spatial relations, maintaining probabilistic semantics while improving extrapolation to unseen primitive combinations. CoGe-GCD is implemented as an inductive-bias module between backbone and projection head, without modifying heads or losses, and can be plugged into diverse GCD frameworks. On standard benchmarks, it consistently improves all-class accuracy, unknown-class number estimation, and geometric quality, with marginal computational overhead. Code is available at https://github.com/lytang63/CoGe-GCD.

Luyao Tang, Jiewei Zheng, Kunze Huang +3

The University of Hong Kong, Hong Kong SAR, China · Xiamen University, Xiamen, China · Shenzhen University, Shenzhen, China.

Geometric Representation Learning Classification Compositional Generalization Generalized Category Discovery