Spectral Flow on the Manifold of SPD Matrices for Multimodal Data Processing
This addresses multimodal data integration for applications like sensor fusion, though it appears incremental as it builds on existing manifold learning and Riemannian geometry.
The paper tackles the problem of processing multimodal data with shared and measurement-specific variability by analyzing spectral changes along geodesics on the SPD matrix manifold, enabling unsupervised extraction of common and specific latent components.
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.