An adaptive stochastic approximation algorithm for simultaneous diagonalization of matrix sequences with applications

Abstract

We describe an adaptive algorithm based on stochastic approximation theory for the simultaneous diagonalization of the expectations of two random matrix sequences. Although there are several conventional approaches to solving this problem, there are many applications in pattern analysis and signal detection that require an online (i.e., real-time) procedure for this computation. In these applications, we are given two sequences of random matrices {A/sub k/} and {B/sub k/} as online observations, with lim/sub k/spl rarr//spl infin//E[A/sub k/]=A and lim/sub k/spl rarr//spl infin//E[B/sub k/]=B, where A and B are real, symmetric and positive definite. For every sample (A/sub k/, B/sub k/), we need the current estimates /spl Phi//sub k/ and /spl Lambda//sub k/, respectively of the eigenvectors /spl Phi/ and eigenvalues /spl Lambda/ of A/sup -1/ B. We have described such an algorithm where /spl Phi//sub k/ and /spl Lambda//sub k/ converge provably with probability one to /spl Phi/ and /spl Lambda/ respectively. A novel computational procedure used in the algorithm is the adaptive computation of A/sup - 1/2 /. Besides its use in the generalized eigen-decomposition problem, this procedure can be used on its own in several feature extraction problems. The performance of the algorithm is demonstrated with an example of detecting a high-dimensional signal in the presence of interference and noise, in a digital mobile communications problem. Experiments comparing computational complexity and performance demonstrate the effectiveness of the algorithm in this real-time application.