Low-rank Adaptive Filter Research Articles

Theoretical Performance of Low Rank Adaptive Filters in the Large Dimensional Regime

This paper proposes a new approximation of the theoretical signal to interference plus noise ratio (SINR) loss of the low-rank (LR) adaptive filter built on the eigenvalue decomposition of the sample covariance matrix. This new result is based on an analysis in the large dimensional regime, i.e., when the size and the number of data tend to infinity at the same rate. Compared to previous works, this new derivation allows us to measure the quality of the adaptive filter near the LR contribution. Moreover, we propose a new LR adaptive filter and we also derive its SINR loss approximation in a large dimensional regime. We validate these results on a jamming application and test their robustness in a multiple input multiple output space time adaptive processing application where the data size is large.

On the Asymptotics of Maronna's Robust PCA

The eigenvalue decomposition (EVD) parameters of the second order statistics are ubiquitous in statistical analysis and signal processing. Notably, the EVD of the M-estimators of the scatter matrix is a popular choice to perform robust probabilistic PCA or other dimension reduction related applications. Towards the goal of characterizing this process, this paper derives new asymptotics for the EVD parameters (i.e. eigenvalues, eigenvectors, and principal subspace) of M-estimators in the context of complex elliptically symmetric distributions. First, their Gaussian asymptotic distribution is obtained by extending standard results on the sample covariance matrix in the Gaussian context. Second, their convergence towards the EVD parameters of a Gaussian-Core Wishart Equivalent is derived. This second result represents the main contributioninthe sense that it quantifies when itis acceptable to directly rely on well-established results on the EVD of Wishartdistributed matrix for characterizing the EVD of M-estimators. Finally, some examples (intrinsic bias analysis, rank estimation, and low-rank adaptive filtering) illustrate where the obtained results can be leveraged.

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

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/sup 2/), yet they have performance comparable with algorithms having O(N/sup 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.

Low-rank adaptive filters

We introduce a class of adaptive filters based on sequential adaptive eigendecomposition (subspace tracking) of the data covariance matrix. These new algorithms are completely rank revealing, and hence, they can perfectly handle the following two relevant data cases where conventional recursive least squares (RLS) methods fail to provide satisfactory results: (1) highly oversampled smooth data with rank deficient of almost rank deficient covariance matrix and (2) noise-corrupted data where a signal must be separated effectively from superimposed noise. This paper contradicts the widely held belief that rank revealing algorithms must be computationally more demanding than conventional recursive least squares. A spatial RLS adaptive filter has a complexity of O(N/sup 2/) operations per time step, where N is the filter order. The corresponding low-rank adaptive filter requires only O(Nr) operations per time step, where r/spl les/N denotes the rank of the data covariance matrix. Thus, low-rank adaptive filters can be computationally less (or even much less) demanding, depending on the order/rank ratio N/r or the compressibility of the signal. Simulation results substantiate our claims. This paper is devoted to the theory and application of fast orthogonal iteration and bi-iteration subspace tracking algorithms.