Learning Log-Determinant Divergences for Positive Definite Matrices

TL;DR

This paper introduces a data-driven approach to learn similarity measures for SPD matrices by optimizing the ndeta-log-det divergence, enhancing performance across various vision tasks.

Contribution

It proposes a novel method to learn parameters of the ndeta-log-det divergence for SPD matrices, increasing flexibility and applicability in machine learning tasks.

Findings

Improved accuracy in SPD matrix clustering and classification.

Effective optimization via Riemannian gradient descent.

Demonstrated benefits on eight standard vision datasets.

Abstract

Representations in the form of Symmetric Positive Definite (SPD) matrices have been popularized in a variety of visual learning applications due to their demonstrated ability to capture rich second-order statistics of visual data. There exist several similarity measures for comparing SPD matrices with documented benefits. However, selecting an appropriate measure for a given problem remains a challenge and in most cases, is the result of a trial-and-error process. In this paper, we propose to learn similarity measures in a data-driven manner. To this end, we capitalize on the \alpha\beta-log-det divergence, which is a meta-divergence parametrized by scalars \alpha and \beta, subsuming a wide family of popular information divergences on SPD matrices for distinct and discrete values of these parameters. Our key idea is to cast these parameters in a continuum and learn them from data. We systematically extend this idea to learn vector-valued parameters, thereby increasing the expressiveness of the underlying non-linear measure. We conjoin the divergence learning problem with several standard tasks in machine learning, including supervised discriminative dictionary learning and unsupervised SPD matrix clustering. We present Riemannian gradient descent schemes for optimizing our formulations efficiently, and show the usefulness of our method on eight standard computer vision tasks.