Geometry-aware Similarity Learning on SPD Manifolds for Visual Recognition

TL;DR

This paper introduces a geometry-aware SPD similarity learning framework that leverages Riemannian geometry to improve discriminative feature learning on SPD manifolds for visual recognition tasks.

Contribution

It proposes a novel method to optimize over the PSD manifold directly, enhancing SPD feature discrimination for visual recognition.

Findings

Outperforms existing SPD-based discriminant learning methods

Effective on three visual classification tasks

Utilizes Riemannian geometry for optimization

Abstract

Symmetric Positive Definite (SPD) matrices have been widely used for data representation in many visual recognition tasks. The success mainly attributes to learning discriminative SPD matrices with encoding the Riemannian geometry of the underlying SPD manifold. In this paper, we propose a geometry-aware SPD similarity learning (SPDSL) framework to learn discriminative SPD features by directly pursuing manifold-manifold transformation matrix of column full-rank. Specifically, by exploiting the Riemannian geometry of the manifold of fixed-rank Positive Semidefinite (PSD) matrices, we present a new solution to reduce optimizing over the space of column full-rank transformation matrices to optimizing on the PSD manifold which has a well-established Riemannian structure. Under this solution, we exploit a new supervised SPD similarity learning technique to learn the transformation by regressing the similarities of selected SPD data pairs to their ground-truth similarities on the target SPD manifold. To optimize the proposed objective function, we further derive an algorithm on the PSD manifold. Evaluations on three visual classification tasks show the advantages of the proposed approach over the existing SPD-based discriminant learning methods.