Approximate joint matrix diagonalization by Riemannian-gradient-based optimization over the unitary group (with application to neural multichannel blind deconvolution)

The aim of joint matrix diagonalization is to simultaneously diagonalize a set ofmatrices via a similarity transformation. Exact joint diagonalization may be achievedfor no more than two matrices, so when the number of the matrices is larger than two,the joint diagonalization problem becomes an optimization problem which gives riseto an approximate joint diagonalization method. The present chapter deals with theproblem of joint complex-valued matrix diagonalization in terms of a cost function tobe minimized by a unitary transformation with no constraints of symmetry on the setof matrices to be approximately diagonalized. In particular, we propose to solve suchoptimization problem through a Riemannian-gradient-based stepping method over theunitary group of matrices endowed with a method to define a stepsize schedule to beevaluated automatically at every iteration. Also, we describe an application of approximatejoint matrix diagonalization to multichannel blind deconvolution by neuralnetworks in the frequency-domain. To evaluate the performances of the proposed jointdiagonalization method in the context of neural multichannel blind deconvolution, wecompare the performances of the proposed method with the performances exhibitedby two alternativemethods, namely, the JADE algorithmand the direct-search method. © 2010 by Nova Science Publishers, Inc. All rights reserved.