Fusing Eigenvalues

Abstract

In this paper, we propose a new regularized (penalized) covariance matrix estimator which encourages grouping of the eigenvalues by penalizing large differences (gaps) between successive eigenvalues. This is referred to as fusing eigenvalues (eFusion), The proposed penalty function utilizes Tukey's biweight function that is widely used in robust statistics. The main advantage of the proposed method is that it has very small bias for sufficiently large values of penalty parameter. Hence, the method provides accurate grouping of eigenvalues. Such benefits of the proposed method are illustrated with a numerical example, where the method is shown to perform favorably compared to a state-of-art method.

Spectral Shrinkage of Tyler's $M$-Estimator of Covariance Matrix

“…In this scenario, one can leverage on recent eigenvalues-fusing algorithms [14,15] to get a first estimate of the spectral structure (rank and plateaus). Figure 3 illustrates the performance ofΣ SRTy using eFusion [15] (with corresponding estimator denotedΣ eFusion ) to build its target spectrum, that is, the target spectrum was chosen as Λ T = φ(Σ eFusion ). On the left panel of Figure 3, the samples are close to Gaussian (namely, the degrees of freedom of the t-distribution is d = 20).…”

“…In many applications, covariance matrices exhibit particular spectral structures [12][13][14][15], such as low-rank ones. Exploiting this structure through regularized M -estimators requires an appropriate setting of the target T, that raises the following dilemma: i) using T = I (or cI with c an estimate of the mean of the eigenvalues) shrinks the eigenvalues of the estimate toward identity (or grand mean), which does not reflect most spectral structure; ii) using another arbitrary matrix (e.g., diagonal) can account for a specific spectral structure, but also shrinks inherently the eigenvectors of the estimate, which may not be desired in general.…”

“…A majorization-minimization (MM) algorithm [17] is derived to compute this estimator, and the g-convexity [18] of the objective is also discussed. Several simulations illustrate the interest of the approach and also explore a method to automatically select the target spectrum with the eFusion algorithm [15].…”