Blind deconvolution of covariance matrix inverses for autoregressive processes

TL;DR

This paper investigates the problem of blind matrix deconvolution for matrices related to autoregressive processes, providing full characterizations for orders one and two, and limitations for higher orders.

Contribution

It offers a complete characterization of when blind deconvolution is possible for inverse autocovariance matrices of AR processes, including construction methods and theoretical limitations.

Findings

Deconvolution is possible for AR(1) and AR(2) models under certain parameters.

Explicit construction schemes are provided for AR(1) and AR(2) cases.

Deconvolution does not exist for higher-order models when certain conditions are met.

Abstract

Matrix $\mathbf{C}$ can be blindly deconvoluted if there exist matrices $\mathbf{A}$ and $\mathbf{B}$ such that $\mathbf{C}= \mathbf{A} \ast \mathbf{B}$, where $\ast$ denotes the operation of matrix convolution. We study the problem of matrix deconvolution in the case where matrix $\mathbf{C}$ is proportional to the inverse of the autocovariance matrix of an autoregressive process. We show that the deconvolution of such matrices is important in problems of Hankel structured low-rank approximation (HSLRA). In the cases of autoregressive models of orders one and two, we fully characterize the range of parameters where such deconvolution can be performed and provide construction schemes for performing deconvolutions. We also consider general autoregressive models of order $p$, where we prove that the deconvolution $\mathbf{C}= \mathbf{A} \ast \mathbf{B}$ does not exist if the matrix $\mathbf{B}$ is diagonal and its size is larger than $p$.