Geometric Representation Learning

A prevailing paradigm in modern representation learning is the map-first approach, in which a representation map is learned from reconstruction, embedding, or task objectives. At the optimum, when the learned map accurately recovers a global coordinate chart, it should exhibit three structural properties whose geometric meaning can be illustrated through a face-editing example: Commutativity requires that changing pose and then expression gives the same result as applying them in the reverse order; Time Coherence requires that the same variation along one coordinate induces the same expression change across faces; Common-Reference requires that all faces are organized relative to a common reference face. However, small approximation errors in the learned map need not translate into small errors in these structural properties, and can therefore disrupt the global organization of the representation. Based on this observation, we consider the converse of the map-first formulation and ask whether a global representation can instead emerge by directly learning these properties. We represent variations along individual coordinates through vector fields defined in the ambient space and introduce a non-contraction condition preventing one transformation from destroying directions associated with the others. We derive an unsupervised objective that learns these structural properties and establish theoretical results connecting its minimization to tangent-space recovery. Experiments on controlled manifolds validate the predicted tangent-space recovery and global structure, while an autoencoder baseline shows that small map-first errors can still produce substantial violations of the targeted properties.

Representational similarity methods compare the geometries of neural representations, but they do not measure how consistently the geometry of a single representation is recovered from subsets of its feature coordinates. We call this property geometric stability and introduce Shesha, which estimates it by correlating representational dissimilarity matrices from complementary random feature subsets. Shesha is not invariant to orthogonal rotations: representations with identical Gram matrices, and therefore identical linear CKA, can have different geometric stability. Controlled transformations further separate the quantities. Across $2{,}463$ encoder configurations spanning seven domains, similarity and stability are positively associated across non-PCA transformations ($ρ=+0.75$) but negatively associated under PCA-coordinate compression ($ρ=-0.47$). We further evaluate 170 pretrained vision models across six datasets. DINOv2 combines strong transfer performance with bottom-quartile stability on five of six datasets, showing that transferability and feature-split stability need not coincide. Across random feature subsets, the marginal relationship between Shesha and linear-probe variability is dataset-dependent; after controlling for task alignment with LogME, higher Shesha is associated with lower variability on five of six datasets. These results identify geometric stability as a basis-dependent property that complements representational similarity and task alignment.

While Transformers have demonstrated remarkable potential in modeling Partial Differential Equations (PDEs), modeling large-scale unstructured meshes with complex geometries remains a significant challenge. Existing efficient architectures often employ feature dimensionality reduction strategies, which inadvertently induces Geometric Aliasing, resulting in the loss of critical physical boundary information. To address this, we propose the Physics-Geometry Operator Transformer (PGOT), designed to reconstruct physical feature learning through explicit geometry awareness. Specifically, we propose Spectrum-Preserving Geometric Attention (SpecGeo-Attention). Utilizing a ``physics slicing-geometry injection" mechanism, this module incorporates multi-scale geometric encodings to explicitly preserve multi-scale geometric features while maintaining linear computational complexity $O(N)$. Furthermore, PGOT dynamically routes computations to low-order linear paths for smooth regions and high-order non-linear paths for shock waves and discontinuities based on spatial coordinates, enabling spatially adaptive and high-precision physical field modeling. PGOT achieves consistent state-of-the-art performance across four standard benchmarks and excels in large-scale industrial tasks including airfoil and car designs.

Pretrained transformers provide rich, general-purpose embeddings, which are transferred to downstream tasks. However, current transfer strategies: fine-tuning and probing, either distort the pretrained geometric structure of the embeddings or lack sufficient expressivity to capture task-relevant signals. These issues become even more pronounced when supervised data are scarce. Here, we introduce Freeze, Diffuse, Decode (FDD), a novel diffusion-based framework that adapts pre-trained embeddings to downstream tasks while preserving their underlying geometric structure. FDD propagates supervised signal along the intrinsic manifold of frozen embeddings, enabling a geometry-aware adaptation of the embedding space. Applied to antimicrobial peptide design, FDD yields low-dimensional, predictive, and interpretable representations that support property prediction, retrieval, and latent-space interpolation.

Cross-subject motor imagery decoding remains a fundamental challenge in EEG-based brain-computer interfaces due to substantial inter-subject variability. Recent approaches have leveraged Riemannian geometry by representing EEG signals as covariance matrices on the symmetric positive definite (SPD) manifold. However, existing methods primarily focus on manifold-based representations while largely overlooking subject-specific variations in covariance dispersion and orientation. In this work, we address these challenges through geometry-aware congruence transformations and propose three complementary models: (i) Discriminative Congruence Transform (DCT), (ii) Deep Linear DCT (DLDCT), and (iii) Deep DCT-UNet (DDCT-UNet). The proposed models are evaluated both as manifold alignment modules for downstream classifiers and as end-to-end discriminative architectures optimized via cross-entropy with a custom logistic regression head. Experiments on challenging cross-subject motor imagery benchmarks demonstrate consistent improvements in transductive decoding performance, achieving 2-3% higher accuracy than strong baselines. These results highlight the effectiveness of geometry-aware congruence learning for mitigating inter-subject variability in EEG decoding.

We study latent geometry as an explicit component of representation quality in data-scarce learning. For an encoder (φ), we define (Q_{β,γ}(φ)=I(φ(X);Y)-β\mathcal C(φ)-γd_{\mathrm{int}}(φ)), combining task-relevant information with penalties for curvature and intrinsic latent dimension. Thus geometry becomes part of the bottleneck criterion, not only a post hoc diagnostic. Under smooth-manifold, loss-transfer, and estimator-concentration assumptions, we derive non-asymptotic low-label generalization bounds where intrinsic dimension and covering complexity enter explicitly. We characterize the information--geometry frontier and prove empirical-surrogate consistency. The analysis links encoder geometry to learning through latent covering numbers, loss-class entropy, and uniform deviation. We instantiate the theory as \texttt{V-GIB}, adding curvature and dimension penalties to variational bottleneck training. Real low-label benchmarks compare \texttt{V-GIB} with ERM, VIB, and ablations across (1%)--(20%) label fractions. Results show improved performance and reduced geometric complexity in several regimes, especially FashionMNIST and CIFAR-10, while confirming that no fixed regularizer is universally dominant.

Geometry-grounded vision models, such as VGGT, have emerged as robust visual encoders, providing essential geometric priors for robotic manipulation. However, the high computational cost of these models often leads to slow inference, limiting their practical applications in real-world robotics. This paper introduces eVGGT, a lightweight geometry-aware vision encoder distilled from the high-performing VGGT. Our findings demonstrate two primary advantages: i) integrating eVGGT into imitation learning frameworks (including ACT and Diffusion Policy) yields up to a 6.3% improvement in success rate over standard 2D encoders across bimanual and single-arm tasks in both simulation and real-world settings with variable viewpoints; ii) eVGGT achieves a nearly 5 times speedup and a 63% reduction in memory usage compared to state-of-the-art geometry-aware encoders while maintaining comparable task performance. These results suggest that eVGGT substantially alleviates the performance-latency bottleneck that has limited geometry-aware visuomotor policies in real-world deployment.

Detecting partial extrinsic symmetry in 3D geometry is a fundamental yet persistent challenge in computer vision and graphics, critical for tasks ranging from shape completion to procedural generation. Classical transformation-space voting methods rely on pairwise matching, scaling as O(n^2) and struggling to resolve coherent multi-instance groups. Recent learning approaches advance global symmetry detection but restrict the solution space to reflection planes, failing to capture rotational or translational repetitions such as the legs of a chair or the steps of a staircase. We propose SymCL, a self-supervised contrastive learning framework that detects partial symmetries across rotation, translation, and reflection (with scale-invariant features) and requires no ground truth annotations. By mapping local geodesic patches to a latent space invariant to the Euclidean group, we reformulate symmetry detection as a density-based clustering problem, enabling the simultaneous discovery of multi-instance symmetric relationships in a single forward pass. We evaluate quantitatively on SymPartNet, a new benchmark annotating all PartNet categories with partial symmetry relations, and demonstrate class-agnostic generalization qualitatively on everyday objects outside the training distribution.