An Efficient Algorithm for Maximum-Entropy Extension of Block-Circulant Covariance Matrices

TL;DR

This paper introduces an efficient algorithm for completing block-circulant covariance matrices with maximum entropy, improving computational speed and leveraging connections to band-extension problems in signal processing applications.

Contribution

The paper presents a novel, efficient algorithm for maximum-entropy completion of block-circulant matrices, advancing computational methods in covariance matrix extension.

Findings

Algorithm outperforms existing methods in speed and accuracy

Leverages relationship with block-Toeplitz band-extension problems

Applicable to stationary reciprocal process modeling

Abstract

This paper deals with maximum entropy completion of partially specified block-circulant matrices. Since positive definite symmetric circulants happen to be covariance matrices of stationary periodic processes, in particular of stationary reciprocal processes, this problem has applications in signal processing, in particular to image modeling. In fact it is strictly related to maximum likelihood estimation of bilateral AR-type representations of acausal signals subject to certain conditional independence constraints. The maximum entropy completion problem for block-circulant matrices has recently been solved by the authors, although leaving open the problem of an efficient computation of the solution. In this paper, we provide an effcient algorithm for computing its solution which compares very favourably with existing algorithms designed for positive definite matrix extension problems. The proposed algorithm benefits from the analysis of the relationship between our problem and the band-extension problem for block-Toeplitz matrices also developed in this paper.