Spectral Flow on the Manifold of SPD Matrices for Multimodal Data Processing

TL;DR

This paper introduces a novel spectral analysis method on the Riemannian manifold of SPD matrices to improve multimodal data processing, enabling unsupervised extraction of shared and specific features.

Contribution

It combines manifold learning with Riemannian geometry of SPD matrices to analyze spectral changes along geodesics, providing new algorithms for component extraction.

Findings

Effective unsupervised identification of shared and measurement-specific components

Spectral analysis on SPD manifolds reveals underlying data relations

New algorithms outperform traditional methods in multimodal data analysis

Abstract

In this paper, we consider data acquired by multimodal sensors capturing complementary aspects and features of a measured phenomenon. We focus on a scenario in which the measurements share mutual sources of variability but might also be contaminated by other measurement-specific sources such as interferences or noise. Our approach combines manifold learning, which is a class of nonlinear data-driven dimension reduction methods, with the well-known Riemannian geometry of symmetric and positive-definite (SPD) matrices. Manifold learning typically includes the spectral analysis of a kernel built from the measurements. Here, we take a different approach, utilizing the Riemannian geometry of the kernels. In particular, we study the way the spectrum of the kernels changes along geodesic paths on the manifold of SPD matrices. We show that this change enables us, in a purely unsupervised manner, to derive a compact, yet informative, description of the relations between the measurements, in terms of their underlying components. Based on this result, we present new algorithms for extracting the common latent components and for identifying common and measurement-specific components.