Performance analysis of an adaptive algorithm for tracking dominant subspaces

Abstract

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 W W to an orthonormal basis of a dominant invariant subspace of a correlation matrix. We consider the stochastic algorithm governing the evolution of W W WW W W H 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 constrain W W W 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.

Subspace Tracking for Signal Processing

“…W * Fro = O(e −(λr−λr+1)t ). A performance analysis has been given in [24], [25]. This issue will be used as an example analysis of convergence and performance in Subsection VII-C2.…”

Performance Bounds and Statistical Analysis of DOA Estimation

“…. Using (68) with (16) and 23 hkhl Re a H (θ l )Ua(θ k ) (a ′ H (θ k )Π x a ′ (θ l )) and compactly by…”

“…Note that these expressions have been derived in [80] by much more involved derivations based on the asymptotic distribution of the eigenvectors of the sample covariance matrix R x,N . Finally, note that if the sample orthogonal noise projector Π x,N is replaced by an adaptive estimator Π x,γ of Π x , where γ is the step-size of an arbitrary constant step-size recursive stochastic algorithm (see e.g., [16] and [17]), it has been proved in [16] that √ γ( θ γ − θ) converges in distribution to the zero-mean Gaussian distribution of covariance matrix given also by R MUSIC θ , where θ γ is an adaptive estimate of θ given by the MUSIC algorithm based on the specific adaptive estimate Π x,γ of Π x studied in [16].…”

A Bayesian approach to geometric subspace estimation

“…Diffusions have frequently been used to seek global optimizers [2], [20] or to sample from given densities [19], [21], [27] in several applications. With the advent of high-speed desktop computing, such stochastic searches on high-dimensional spaces have become prominent.…”