Regularization and Interpolation of Positive Matrices

TL;DR

This paper develops a new geometric framework for interpolating positive definite matrices, balancing eigenstructure alignment and eigenvalue scaling, to improve spectral analysis of multivariate time series.

Contribution

It introduces a matricial analogue of optimal mass transport tailored for positive definite matrices, addressing artifacts in spectral interpolation.

Findings

New interpolation method reduces push-pop artifacts in spectral analysis.

Framework effectively balances eigenstructure alignment and eigenvalue scaling.

Applicable to multivariate time series power spectral analysis.

Abstract

We consider certain matricial analogues of optimal mass transport of positive definite matrices of equal trace. The framework is motivated by the need to devise a suitable geometry for interpolating positive definite matrices in ways that allow controlling the apparent tradeoff between "aligning up their eigenstructure" and "scaling the corresponding eigenvalues". Indeed, motivation for this work is provided by power spectral analysis of multivariate time series where, linear interpolation between matrix-valued power spectra generates push-pop artifacts. Push-pop of power distribuion is objectionable as it corresponds to unrealistic response of scatterers.