Block Coordinate Descent Algorithms for Auxiliary-Function-Based Independent Vector Extraction

Abstract

In this paper, we address the problem of extracting all super-Gaussian source signals from a linear mixture in which (i) the number of super-Gaussian sources <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> is less than that of sensors <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> , and (ii) there are up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M - K$</tex-math></inline-formula> stationary Gaussian noises that do not need to be extracted. To solve this problem, independent vector extraction (IVE) using a majorization minimization and block coordinate descent (BCD) algorithms has been developed, attaining robust source extraction and low computational cost. We here improve the conventional BCDs for IVE by carefully exploiting the stationarity of the Gaussian noise components. We also newly develop a BCD for a semiblind IVE in which the transfer functions for several super-Gaussian sources are given a priori. Both algorithms consist of a closed-form formula and a generalized eigenvalue decomposition. In a numerical experiment of extracting speech signals from noisy mixtures, we show that when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K = 1$</tex-math></inline-formula> in a blind case or at least <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K - 1$</tex-math></inline-formula> transfer functions are given in a semiblind case, the convergence of our proposed BCDs is significantly faster than those of the conventional ones.