Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods

TL;DR

This paper introduces geometry-aware dimensionality reduction algorithms for high-dimensional SPD matrices, enabling efficient and discriminative representations for visual recognition tasks while reducing computational costs.

Contribution

It proposes a novel method to map high-dimensional SPD matrices to lower-dimensional manifolds using orthonormal projections, formulated as an optimization on Grassmann manifolds.

Findings

Significant accuracy improvements over state-of-the-art methods.

Effective dimensionality reduction with preserved discriminative power.

Fast solutions for special cases of the optimization problem.

Abstract

Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices -especially of high-dimensional ones- comes at a high cost that limits the applicability of existing techniques. In this paper, we introduce algorithms able to handle high-dimensional SPD matrices by constructing a lower-dimensional SPD manifold. To this end, we propose to model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. This lets us formulate dimensionality reduction as the problem of finding a projection that yields a low-dimensional manifold either with maximum discriminative power in the supervised scenario, or with maximum variance of the data in the unsupervised one. We show that learning can be expressed as an optimization problem on a Grassmann manifold and discuss fast solutions for special cases. Our evaluation on several classification tasks evidences that our approach leads to a significant accuracy gain over state-of-the-art methods.