Dimensionality Reduction on Riemannian Manifolds in Data Analysis

In this work, we investigate Riemannian geometry based dimensionality reduction methods that respect the underlying manifold structure of the data. In particular, we focus on Principal Geodesic Analysis (PGA) as a nonlinear generalization of PCA for manifold valued data, and extend discriminant analysis through Riemannian adaptations of other known dimensionality reduction methods. These approaches exploit geodesic distances, tangent space representations, and intrinsic statistical measures to achieve more faithful low dimensional embeddings. We also discuss related manifold learning techniques and highlight their theoretical foundations and practical advantages. Experimental results on representative datasets demonstrate that Riemannian methods provide improved representation quality and classification performance compared to their Euclidean counterparts, especially for data constrained to curved spaces such as hyperspheres and symmetric positive definite manifolds. This study underscores the importance of geometry aware dimensionality reduction in modern machine learning and data science applications.

💡 Research Summary

This paper presents a comprehensive framework for dimensionality reduction and discriminant analysis of data that reside on non‑Euclidean spaces such as spheres, symmetric positive‑definite (SPD) matrices, and Grassmann manifolds. After reviewing the necessary Riemannian geometry—manifolds, tangent spaces, metrics, geodesic distances, exponential and logarithmic maps—the authors introduce the essential tools of Riemannian optimization, including the Riemannian gradient, retractions, and convergence guarantees.

The core contribution is the systematic extension of several classic Euclidean techniques to the intrinsic geometry of manifolds. Principal Geodesic Analysis (PGA) serves as the manifold counterpart of PCA: the Fréchet mean of the data is computed, each sample is mapped to the tangent space via the logarithmic map, and a standard PCA is performed there. The principal directions are then lifted back to the manifold with the exponential map, yielding geodesic components that capture the dominant variation while respecting curvature. Building on PGA, the authors propose Riemannian Robust PCA (RRPCA), which combines low‑rank approximation in the tangent space with a sparse error term to handle outliers.

Orthogonal Neighborhood Preserving Projection (ONPP) is adapted to manifolds by learning a linear projection in the tangent space that preserves local distances; specialized implementations for SPD and Grassmann manifolds exploit matrix logarithms and exponentials for efficient computation. A Riemannian version of Laplacian Eigenmaps constructs a graph using geodesic distances, computes the Laplacian eigenvectors in the tangent space, and obtains a nonlinear embedding that respects the manifold’s intrinsic geometry.

Supervised extensions include Riemannian Linear Discriminant Analysis (RLDA), where between‑class and within‑class scatter are defined via Fréchet means and tangent‑space covariances, and a generalized Rayleigh quotient yields discriminative directions. The paper also introduces Riemannian Isomap, which computes all‑pair geodesic distances and applies classical multidimensional scaling, and a Riemannian Support Vector Machine (RSVM) that employs kernels based on geodesic distances (e.g., log‑Euclidean kernel for SPD data).

Experimental validation is carried out on three representative domains: face images on the unit sphere, texture classification using SPD covariance descriptors, and action recognition with subspace (Grassmann) representations. Across all tasks, the manifold‑aware methods consistently outperform Euclidean baselines (PCA, LDA, Isomap) by 3–7 % in classification accuracy, and RRPCA demonstrates robustness to up to 30 % corrupted samples. Although the additional logarithm/exponential operations increase computational load, the authors report that optimized matrix routines and parallelization keep runtimes practical.

The authors conclude that intrinsic Riemannian approaches provide more faithful low‑dimensional representations for curved data, leading to better downstream performance. They acknowledge challenges such as the cost of Fréchet mean computation, numerical stability of exponential/logarithmic maps, and scalability to large datasets, suggesting these as directions for future work. Overall, the paper offers a valuable synthesis of geometry‑aware dimensionality reduction techniques, positioning them as essential tools for modern machine‑learning applications dealing with non‑Euclidean data.