Robust Geometric Metric Learning

TL;DR

This paper introduces Robust Geometric Metric Learning (RGML), a novel approach that estimates class covariance matrices by shrinking them towards a barycenter using Riemannian geometry, improving robustness and performance.

Contribution

The paper presents RGML, a new metric learning method leveraging Riemannian geometry for covariance estimation, with two specific cost functions, enhancing robustness and accuracy.

Findings

Strong performance on real datasets

Robustness to mislabeled data

Effective covariance estimation using Riemannian geometry

Abstract

This paper proposes new algorithms for the metric learning problem. We start by noticing that several classical metric learning formulations from the literature can be viewed as modified covariance matrix estimation problems. Leveraging this point of view, a general approach, called Robust Geometric Metric Learning (RGML), is then studied. This method aims at simultaneously estimating the covariance matrix of each class while shrinking them towards their (unknown) barycenter. We focus on two specific costs functions: one associated with the Gaussian likelihood (RGML Gaussian), and one with Tyler's M -estimator (RGML Tyler). In both, the barycenter is defined with the Riemannian distance, which enjoys nice properties of geodesic convexity and affine invariance. The optimization is performed using the Riemannian geometry of symmetric positive definite matrices and its submanifold of unit determinant. Finally, the performance of RGML is asserted on real datasets. Strong performance is exhibited while being robust to mislabeled data.