Spatiotemporal Analysis Using Riemannian Composition of Diffusion Operators

TL;DR

This paper introduces Riemannian multi-resolution analysis (RMRA), a novel operator-based method for spatiotemporal analysis of multivariate time-series, leveraging geometry, Riemannian metrics, and spectral analysis to extract dynamic modes.

Contribution

It combines manifold learning, Riemannian geometry, and spectral analysis into a unified framework for analyzing multivariate time-series with geometric structure.

Findings

Theoretical results on spectral properties of composite operators.

Demonstrated effectiveness on simulations.

Validated on real-world data.

Abstract

Multivariate time-series have become abundant in recent years, as many data-acquisition systems record information through multiple sensors simultaneously. In this paper, we assume the variables pertain to some geometry and present an operator-based approach for spatiotemporal analysis. Our approach combines three components that are often considered separately: (i) manifold learning for building operators representing the geometry of the variables, (ii) Riemannian geometry of symmetric positive-definite matrices for multiscale composition of operators corresponding to different time samples, and (iii) spectral analysis of the composite operators for extracting different dynamic modes. We propose a method that is analogous to the classical wavelet analysis, which we term Riemannian multi-resolution analysis (RMRA). We provide some theoretical results on the spectral analysis of the composite operators, and we demonstrate the proposed method on simulations and on real data.