AN ADAPTIVE STOCHASTIC-APPROXIMATION ALGORITHM FOR SIMULTANEOUS DIAGONALIZATION OF MATRIX SEQUENCES WITH APPLICATIONS

We describe an adaptive algorithm based on stochastic approximation th eory for the simultaneous diagonalization of the expectations of two r andom matrix sequences. Although there are several conventional approa ches to solving this problem, there are many applications in pattern a nalysis and signal detection that require an online (i.e., real-time) procedure for this computation. In these applications, we are given tw o sequences of random matrices {A(k)} and {B-k} as online observations , with lim(k-->infinity)E[A(k)] = A and lim(k-->infinity)E[B-k] = B, w here A and B are real, symmetric and positive definite. For every samp le (A(k),B-k), we need the current estimates Phi(k) and Lambda(k) resp ectively of the eigenvectors Phi and eigenvalues Lambda of A(-1) B. We have described such an algorithm where Phi(k) and Lambda(k) converge provably with probability one to Phi and Lambda respectively. A novel computational procedure used in the algorithm is the adaptive computat ion of A(-1/2). Besides its use in the generalized eigen-decomposition problem, this procedure can be used on its own in several feature ext raction problems. The performance of the algorithm is demonstrated wit h an example of detecting a high-dimensional signal in the presence of interference and noise, in a digital mobile communications problem. E xperiments comparing computational complexity and performance demonstr ate the effectiveness of the algorithm in this real-time application.