Performance analysis of an adaptive algorithm for tracking dominant subspaces

This paper provides a performance analysis of a least mean square (LMS) dominant invariant subspace algorithm. Based on an unconstrained minimization problem, this algorithm is a stochastic gradient algorithm driving the columns of a matrix W to an orthonormal basis of a dominant invariant subspace of a correlation matrix. We consider the stochastic algorithm governing the evolution of WW to the projection matrix onto this dominant invariant subspace and study its asymptotic distribution. A closed-form expression of its asymptotic covariance is given in the case of independent observations and is further analyzed to provide some insights into the behavior of this LMS type algorithm. In particular, it is shown that even though the algorithm does not constrainW to have orthonormal columns, there is deviation from orthonormality of the first order. We also give a closed-form expression of the asymptotic covariance of DOA’s estimated by the MUSIC algorithm applied to the adaptive estimate of the projector. It is found that the asymptotic distributions have a structure that is very similar to those describing batch estimation techniques because both algorithms are obtained from the minimization of the same criterion. Finally, the accuracy of the asymptotic analysis is checked by numerical simulations and is found to be valid not only for a “small” step size but in a very large domain.