Independent Vector Analysis via Log-Quadratically Penalized Quadratic Minimization

Abstract

We propose a new algorithm for blind source separation of convolutive mixtures using independent vector analysis. This is an improvement over the popular auxiliary function based independent vector analysis (AuxIVA) with iterative projection (IP) or iterative source steering (ISS). We introduce iterative projection with adjustment (IPA), whereas we update one demixing filter and jointly adjust all the other sources along its current direction. We implement this scheme as multiplicative updates by a rank-2 perturbation of the identity matrix. Each update involves solving a non-convex minimization problem that we term log-quadratically penalized quadratic minimization (LQPQM), that we think is of interest beyond this work. We find that the global minimum of an LQPQM can be efficiently computed. In the general case, we show that all its stationary points can be characterized as zeros of a kind of secular equation, reminiscent of modified eigenvalue problems. We further prove that the global minimum corresponds to the largest of these zeros. We propose a simple procedure based on Newton-Raphson seeded with a good initial point to efficiently compute it. We validate the performance of the proposed method for blind acoustic source separation via numerical experiments with reverberant speech mixtures. We show that not only is the convergence speed faster in terms of iterations, but each update is also computationally cheaper. Notably, for four and five sources, AuxIVA with IPA converges more than twice as fast as competing methods.