Dimensionality Reduction on Riemannian Manifolds in Data Analysis
This work addresses the need for geometry-aware dimensionality reduction in machine learning and data science, particularly for data constrained to curved spaces, but it is incremental as it builds on existing Riemannian adaptations.
The paper tackled the problem of dimensionality reduction for data on Riemannian manifolds by extending methods like Principal Geodesic Analysis and discriminant analysis, resulting in improved representation quality and classification performance compared to Euclidean methods, especially on curved spaces like hyperspheres and symmetric positive definite manifolds.
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.