Riemannian submanifold framework for log-Euclidean metric learning on symmetric positive definite manifolds

Abstract

This study presents a novel Riemannian submanifold (RS) framework for log-Euclidean metric learning on symmetric positive definite manifolds. Our method identifies the optimal RS without changing the original tangent space. The RS is spanned by multiple bases, and each data point is parameterized using these bases, such that the data can be represented more informatively compared to conventional approaches. In the RS, a new distance function is defined, and its derivative cannot be obtained trivially. We overcome this difficulty and provide a simple analytic form of the derivative. The proposed transformation matrix can take any form, which gives flexibility to our method and enables the creation of several variants. In experiments, our method and its variants surpass state-of-the-art metric learning methods in synthetic, material categorization, and action recognition problems.