A Direct Algorithm for Nonorthogonal Approximate Joint Diagonalization

Abstract

While a pair of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> × <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> matrices can almost always be exactly and simultaneously diagonalized by a generalized eigendecomposition, no exact solution exists in the case of a set with more than two matrices. This problem, termed approximate joint diagonalization (AJD), is instrumental in blind signal processing. When the set of matrices to be jointly diagonalized includes at least <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> linearly independent matrices, we propose a suboptimal but closed-form solution for AJD in the direct least-squares sense. The corresponding non-iterative algorithm is given the acronym DIEM (DIagonalization using Equivalent Matrices). Extensive numerical simulations show that DIEM is both fast and accurate compared to the state-of-the-art iterative AJD algorithms.