tangent space mapping

This work addresses the limitations of traditional Euclidean dimensionality reduction methods in effectively handling data intrinsically residing on nonlinear Riemannian manifolds—such as hyperspheres or the manifold of symmetric positive-definite matrices. By extending classical techniques like principal component analysis and discriminant analysis into a Riemannian geometric framework, the study proposes geometry-aware nonlinear dimensionality reduction approaches grounded in geodesic distances, tangent space mappings, and intrinsic statistical measures. These include Principal Geodesic Analysis (PGA) and manifold-based discriminant analysis. Experimental results demonstrate that the proposed methods significantly outperform their Euclidean counterparts on benchmark datasets embedded in curved spaces, achieving superior preservation of intrinsic manifold structure, enhanced quality of low-dimensional embeddings, and improved downstream classification performance.

This paper addresses classification of curve-shaped data by proposing a Riemannian geometric framework for linear and quadratic discriminant analysis (LDA/QDA) on shape manifolds. Methodologically, shapes are represented via the square-root velocity function (SRVF), and the infinite-dimensional shape manifold is locally linearized through tangent-space projection. Riemannian means and covariances are estimated in the tangent space, and dimensionality reduction is achieved using Fourier basis coefficients. The resulting classifier is invariant to translation, rotation, and scaling. The key contribution is the first systematic adaptation of classical LDA/QDA to the tangent plane of shape space, overcoming fundamental limitations of Euclidean approaches in preserving shape invariances. Experiments demonstrate superior classification performance on both synthetic datasets and real biomedical shapes—including cortical sulci, the corpus callosum, and midline facial features in fetal alcohol syndrome.

Existing studies fail to explain why Euclidean classifiers can be directly applied to Riemannian features after matrix power normalization. Method: From a Riemannian geometric perspective, we provide the first unified interpretation of the intrinsic roles of matrix logarithm and power normalization on the Symmetric Positive Definite (SPD) manifold: they are not mere linear mappings but implicitly realize a Riemannian classifier—equivalently performing geodesic distance classification on the manifold via tangent space projection. We establish a rigorous theoretical correspondence between normalization operations and the Riemannian classifier, grounded in covariance pooling modeling, matrix function analysis, and SPD manifold theory. Contribution/Results: Our unified framework ensures theoretical consistency, significantly enhancing both interpretability and performance. Extensive experiments on fine-grained and large-scale visual classification benchmarks validate the theory. The implementation is publicly available.

This work addresses the challenge of designing efficient neural networks on non-compact symmetric spaces, such as hyperbolic space and the manifold of symmetric positive definite (SPD) matrices. The authors propose a unified framework grounded in G-invariant Riemannian metrics and differential geometry, which naturally subsumes existing models as special cases. Central to this framework is the first closed-form solution for the distance between a point and a hyperplane in high-rank non-compact symmetric spaces, enabling the construction of tailored fully connected layers and attention mechanisms. Extensive experiments demonstrate that the proposed architecture achieves significant performance gains across diverse tasks, including image classification, EEG signal processing, image generation, and natural language inference.

This work investigates how training dynamically shapes the Riemannian geometric structure induced by neural network representations. **Problem**: While deep networks operate in high-dimensional feature spaces, the geometric evolution of their induced Riemannian metric during training remains poorly understood. **Method**: We integrate Riemannian geometric analysis, infinite-width network theory, and feature-space metric modeling, conducting systematic experiments across supervised and self-supervised learning paradigms. **Contribution/Results**: We theoretically prove that infinitely wide random networks initially possess an isotropic (highly symmetric) Riemannian metric; training actively breaks this symmetry by locally amplifying the metric tensor near decision boundaries—a phenomenon we term *boundary-sensitive metric amplification*. Empirical validation across deep image classification and self-supervised learning confirms its robustness, revealing a self-emergent geometric inductive bias. This provides a novel geometric paradigm for understanding the intrinsic geometry of nonlinear feature learning.

Existing manifold-based deep learning approaches are often limited to specific manifolds, rely on Euclidean approximations, or involve computationally expensive and numerically unstable geometric operations. This work proposes a unified Riemannian deep learning framework whose core innovations include generalizing batch normalization to Lie and rotation groups, extending multinomial logistic regression to arbitrary Riemannian manifolds, and designing an adaptive, efficient metric for symmetric positive definite (SPD) matrices. The framework integrates generalized batch normalization, Riemannian multinomial regression, unconstrained modeling in hyperbolic space, Busemann function learning, and Cholesky/Log-Euclidean geometries. Theoretical analysis and experiments demonstrate that the proposed method achieves superior performance and computational efficiency across diverse tasks in computer vision, signal processing, graph learning, and genomics.

This study addresses the lack of theoretical guidance in selecting embeddings for symmetric positive definite (SPD) manifold learning by constructing a unified Transformer framework to systematically investigate the impact of Bures–Wasserstein SPD (BWSPD), Log-Euclidean, and Euclidean embeddings on EEG covariance matrix classification. It establishes, for the first time, a theoretical link between SPD embedding geometry and optimization dynamics, proposing BN-Embed as an approximate Riemannian normalization scheme and proving that the isometric property of BWSPD is uniquely determined by the condition number ratio κ. Experimental results demonstrate that the Log-Euclidean Transformer achieves state-of-the-art performance across three EEG paradigms, significantly outperforming conventional Riemannian methods, while BWSPD excels in high-dimensional settings—yielding a 26% accuracy gain over prior approaches in a 56-channel ERP task.

This work addresses the limitations of existing symmetric positive definite (SPD) methods in EEG analysis, which often overlook segment-specific synchronization and local topological structures of brain regions, thereby failing to accurately characterize functional connectivity. To overcome this, the authors propose RepSPD, a novel model that uniquely integrates dynamic graph-driven functional connectivity with SPD manifold representations. The approach leverages a cross-attention mechanism on the Riemannian manifold to modulate the geometric properties of SPD matrices and introduces a global bidirectional alignment strategy to refine tangent space embeddings, effectively mitigating geometric distortions caused by manifold curvature. Extensive experiments demonstrate that RepSPD significantly outperforms state-of-the-art methods across multiple EEG tasks, exhibiting superior robustness and generalization capability.

This work addresses the sensitivity of commonly used similarity measures—such as cosine similarity—in neural representation analysis to coordinate transformations, stemming from their neglect of the intrinsic gauge freedom in representation spaces. Adopting a differential-geometric perspective, the paper conceptualizes neural representations as equivalence classes under the action of the general linear group and introduces, for the first time, the notion of gauge freedom to unify explanations for phenomena like the instability of cosine similarity and embedding anisotropy. The authors argue that analyses should focus on gauge-invariant quantities or explicitly fix a gauge coordinate system. Through experiments with multilayer perceptrons and convolutional networks using methods such as SVCCA and CKA, they demonstrate that inserting invertible linear transformations—while preserving model predictions—can drastically distort similarity and neighborhood structures, thereby revealing the strong dependence of current metrics on the choice of metric (gauge).

This work addresses the problem of learning low-dimensional embeddings from high-dimensional data while preserving the intrinsic geometric structure inherent in self-reconstruction. To this end, the authors propose a novel approach grounded in reproducing kernel Hilbert spaces (RKHS), which explicitly models the self-expressive property of data by integrating the representer theorem with separable operator-valued kernels. Geometric structure preservation is achieved through kernel alignment, enabling effective transfer of intrinsic data geometry into the embedding space. The proposed method unifies manifold learning and kernel methods within a coherent framework. Empirical evaluations on both synthetic benchmarks—such as concentric circles and the Swiss roll—and real-world datasets—including molecular activity prediction in cancer research and intrusion detection in IoT systems—demonstrate its superior performance in maintaining reconstruction-based geometric fidelity.