ISS2: An Extension of Iterative Source Steering Algorithm for Majorization-Minimization-Based Independent Vector Analysis

A majorization-minimization (MM) algorithm for independent vector analysis optimizes a separation matrix
$W:= [w_{1},\ldots,\ w_{m}]^{\mathrm{H}}\in \mathbb{C}^{m\times\overline{m}}$
by minimizing a surrogate function of the form
$\mathcal{L}(W):=\sum_{i=1}^{m}w_{i}^{\mathrm{H}}V_{i}w_{i}-\log\vert \det W\vert ^{2}$
, where
$m\in \mathbb{N}$
is the number of sensors and positive definite matrices
$v_{1}, \ldots,V_{m}\in \mathbb{C}^{m\times m}$
are constructed in each MM iteration. For
$m\geq 3$
, no algorithm has been found to obtain a global minimum of
$\mathcal{L}(W)$
. Instead, block coordinate descent (BCD) methods with closed-form update formulas have been developed for minimizing
$\mathcal{L}(W)$
and shown to be effective. One such BCD is called iterative projection (IP) that updates one or two rows of
$W$
in each iteration. Another BCD is called iterative source steering (ISS) that updates one column of the mixing matrix
$\mathcal{A}:=W^{-1}$
in each iteration. Although the time complexity per iteration of ISS is
$m$
times smaller than that of IP, the conventional ISS converges slower than the current fastest IP (called IP
2
) that updates two rows of
$W$
in each iteration. We here extend this ISS to ISS
2
that can update two columns of
$A$
in each iteration while maintaining its small time complexity. To this end, we provide a unified way for developing new ISS type methods from which ISS
2
as well as the conventional ISS can be immediately obtained in a systematic manner. Numerical experiments to separate reverberant speech mixtures show that our ISS
2
converges in fewer MM iterations than the conventional ISS, and is comparable to IP
2
.