Dimensionality Reduction on Riemannian Manifolds in Data Analysis

TL;DR

The paper addresses dimensionality reduction for data constrained to nonlinear Riemannian manifolds, where standard Euclidean PCA can distort the intrinsic structure. It surveys and develops geometry-aware methods, centering on Principal Geodesic Analysis and Riemannian adaptations of PCA, LDA, Isomap, Laplacian Eigenmaps, and SVM, all leveraging intrinsic geometry. A unifying framework maps data to tangent spaces around the Fréchet mean, performs linear analysis there, and maps results back to the manifold when needed, enabling intrinsic embeddings. Experimental results across real, manifold, and spherical datasets show that Riemannian methods achieve higher representation fidelity and classification accuracy, with especially strong gains on curved spaces such as hyperspheres and SPD manifolds, underscoring the practical value of geometry-aware dimensionality reduction.

Abstract

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.

Figures (8)

• Figure 2.1: Tangent space and geodesics on the sphere $\mathbb{S}^2$. (Left) The tangent space $T_p\mathcal{M}$ at point $p$. (Right) Geodesics emanating from $p$ induced by tangent vectors via the exponential map.
• Figure 2.2: Exponential and logarithmic maps on $\mathbb{S}^2$. Left: $\mathrm{Exp}_p(v)$ maps tangent vector $v$ to point $q$ via the geodesic. Right: $\mathrm{Log}_p(q)$ returns the tangent vector whose exponential gives $q$.
• Figure 11.1: Visualization of 3D manifold and spherical datasets. Spherical datasets (bottom row) exhibit intrinsic geometry on $\mathcal{S}^2$.
• Figure 11.2: 3D embeddings of Breast Cancer dataset.
• Figure 11.3: 3D embeddings of MNIST dataset.

Theorems & Definitions (10)

• Definition 2.1
• Definition 2.2: Tangent Spaces
• Definition 2.3: Riemannian Metric
• remark 1
• Definition 3.1: Differential of a Function
• Theorem 3.2
• Theorem 3.3
• remark 2
• remark 3
• remark 4