Symmetric Positive Semi-definite Riemannian Geometry with Application to Domain Adaptation

TL;DR

This paper advances the Riemannian geometry of SPSD matrices by providing new approximations and formulas, enabling improved domain adaptation techniques demonstrated in hyper-spectral image fusion and motion identification.

Contribution

It introduces new approximations of geodesic-related maps, a closed-form for Parallel Transport, and a canonical form for SPSD matrices, enhancing domain adaptation methods.

Findings

Improved domain adaptation performance in hyper-spectral image fusion.

Effective motion identification using the proposed geometric methods.

New mathematical tools for SPSD matrix analysis.

Abstract

In this paper, we present new results on the Riemannian geometry of symmetric positive semi-definite (SPSD) matrices. First, based on an existing approximation of the geodesic path, we introduce approximations of the logarithmic and exponential maps. Second, we present a closed-form expression for Parallel Transport (PT). Third, we derive a canonical representation for a set of SPSD matrices. Based on these results, we propose an algorithm for Domain Adaptation (DA) and demonstrate its performance in two applications: fusion of hyper-spectral images and motion identification.