Spatiotemporal Analysis Using Riemannian Composition of Diffusion Operators

Abstract

Multivariate time-series have become abundant in recent years, as many dataacquisition systems record information through multiple sensors simultaneously. In this paper, we assume the variables pertain to some geometry and present an operatorbased 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 positivedefinite 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.

ManiFeSt: Manifold-based Feature Selection for Small Data Sets

“…For each kernel, K , the most dominant components of these feature associations are captured by eigenvectors that correspond to the largest eigenvalues. The motivation for our score then comes from the spectral properties of the difference operator, D, which were recently analyzed in [61]. This work proved that the leading eigenvectors of D (corresponding to the largest eigenvalues in absolute value) are related to similar eigenvectors of K 1 and K 2 that correspond to significantly different eigenvalues.…”

“…Following [61], and based on the approximation of the geodesic path in (11), we define the mean and difference operators for two SPSD matrices, K 1 and K 2 , whose structure space representation is given by K ∼ = (G , P ) where G ∈ V d,k and P…”

“…Proof. A proof showing that φ and λ (D) are an approximate eigen-pair of D can be found in [61,Theorem 4]. For the bound on λ (D) , note that equation (15) holds for K 1 with no change, whereas for K 2 we have:…”

“…First, the feature manifold of each class is learned using a symmetric positive-definite (SPD) kernel. Then, based on the Riemannian geometry of SPD matrices [51,9], we build a composite symmetric kernel that captures the differences between the geometries underlying the feature spaces [61]. Specifically, we apply spectral analysis to the composite kernel and propose a score that reveals discriminative features.…”

Log-Euclidean Signatures for Intrinsic Distances Between Unaligned Datasets

“…Following previous work [Coifman and Hirn, 2014;Katz et al, 2020;Shnitzer et al, 2022], we define an SPD matrix that is similar to W DM given by:…”