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

Abstract

A majorization-minimization (MM) algorithm for independent vector analysis optimizes a separation matrix $W = [w_1, \ldots, w_m]^h \in \mathbb{C}^{m \times m}$ by minimizing a surrogate function of the form $\mathcal{L}(W) = \sum_{i = 1}^m w_i^h V_i w_i - \log | \det W |^2$, 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 $A = W^{-1}$ 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 $\text{IP}_2$) that updates two rows of $W$ in each iteration. We here extend this ISS to $\text{ISS}_2$ 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 $\text{ISS}_2$ as well as the conventional ISS can be immediately obtained in a systematic manner. Numerical experiments to separate reverberant speech mixtures show that our $\text{ISS}_2$ converges in fewer MM iterations than the conventional ISS, and is comparable to $\text{IP}_2$.