Interpolation and Extrapolation of Toeplitz Matrices via Optimal Mass Transport

TL;DR

This paper introduces a spectral optimal mass transport-based distance for Toeplitz matrices, enabling structure-preserving interpolation, extrapolation, and noise-robust signal processing applications.

Contribution

It presents a novel spectral domain distance measure for Toeplitz matrices that preserves positive semi-definiteness and structure during interpolation and extrapolation.

Findings

Distance measure is contractive under noise

Method preserves Toeplitz structure during interpolation

Applications include clustering and tracking of stochastic processes

Abstract

In this work, we propose a novel method for quantifying distances between Toeplitz structured covariance matrices. By exploiting the spectral representation of Toeplitz matrices, the proposed distance measure is defined based on an optimal mass transport problem in the spectral domain. This may then be interpreted in the covariance domain, suggesting a natural way of interpolating and extrapolating Toeplitz matrices, such that the positive semi-definiteness and the Toeplitz structure of these matrices are preserved. The proposed distance measure is also shown to be contractive with respect to both additive and multiplicative noise, and thereby allows for a quantification of the decreased distance between signals when these are corrupted by noise. Finally, we illustrate how this approach can be used for several applications in signal processing. In particular, we consider interpolation and extrapolation of Toeplitz matrices, as well as clustering problems and tracking of slowly varying stochastic processes.