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

arXiv: https://arxiv.org/abs/2009.08062 · HTML: https://mlsanity.com/paper/2009.08062/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Sep 2020.

## Built on by (most cited first, 4 of 4)

- [Spatiotemporal Analysis Using Riemannian Composition of Diffusion Operators](https://mlsanity.com/paper/2201.08530/) (2022)
- [Log-Euclidean Signatures for Intrinsic Distances Between Unaligned Datasets](https://mlsanity.com/paper/2202.01671/) (2022)
- [Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy](https://mlsanity.com/paper/2410.21107/) (2024)
- [On Learning what to Learn: heterogeneous observations of dynamics and establishing (possibly causal) relations among them](https://mlsanity.com/paper/2406.06812/) (2024)

## Abstract (arXiv)

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.