Efficient, high performance, subspace tracking for time-domain data

Abstract

This paper describes two new algorithms for tracking the subspace spanned by the principal eigenvectors of the correlation matrix associated with time-domain (i.e. time series) data. The algorithms track the d principal N -dimensional eigenvectors of the data covariance matrix with a complexity of O(Nd 2 ), yet have performance comparable to algorithms having O(N 2 d) complexity. The computation reduction is achieved by exploiting the shift-invariance property of temporal data covariance matrices. Experiments are used to compare our algorithms with other well-known approaches of similar and/or lower complexity. Our new algorithms are shown to outperform the subset of the general approaches having the same complexity. The new algorithms are useful for applications such as subspace-based speech enhancement, and low-rank adaptive filtering.

Fast approximated power iteration subspace tracking

“…Note that if β J −1 + y(t) H h(t) is singular, Z(t) and W (t) can no longer be updated with equations (34) and (37). In practice, we never encountered this rank deficiency case in our numerical simulations 6 .…”

“…• I. Karasalo's algorithm [2], • the Fast Subspace Tracking (FST) algorithm [3], • the novel PAST algorithm employing Householder transformations, herein called Householder PAST [34], • the Low-Rank Adaptive Filter (Loraf2) algorithm [7], • and the Subspace Projection (SP1) algorithm [37]. Figure 2-a shows that the behaviors of FAPI and Karasalo's algorithm are very similar.…”

“…Finally, substituting equation (42) into equation (39) and right multiplying by S(t) −H = Z(t) H yields equation (37).…”

Generalized YAST algorithm for signal subspace tracking

“…Specially, the YAST algorithms have the best convergence rate among all fast subspace tracking algorithms. This incredible performance is due to an optimal approach, first introduced in the SP algorithm [26]. Remember that SP belongs to the medium burden category.…”

“…The YAST algorithm, which is a fast implementation of the SP 5 algorithm [26], can converge even faster than FAPI and Bi-LS3 algorithms. Specially, the YAST algorithms have the best convergence rate among all fast subspace tracking algorithms.…”