Ilya Soloveychik

Tyler's Covariance Matrix Estimator in Elliptical Models With Convex Structure

We address structured covariance estimation in elliptical distributions by assuming that the covariance is a priori known to belong to a given convex set, e.g., the set of Toeplitz or banded matrices. We consider the General Method of Moments (GMM) optimization applied to robust Tylers scatter M-estimator subject to these convex constraints. Unfortunately, GMM turns out to be non-convex due to the objective. Instead, we propose a new COCA estimator-a convex relaxation which can be efficiently solved. We prove that the relaxation is tight in the unconstrained case for a finite number of samples, and in the constrained case asymptotically. We then illustrate the advantages of COCA in synthetic simulations with structured compound Gaussian distributions. In these examples, COCA outperforms competing methods such as Tylers estimator and its projection onto the structure set.

Group Symmetric Robust Covariance Estimation

In this paper, we consider Tylers robust covariance M-estimator under group symmetry constraints. We assume that the covariance matrix is invariant to the conjugation action of a unitary matrix group, referred to as group symmetry. Examples of group symmetric structures include circulant, perHermitian, and proper quaternion matrices. We introduce a group symmetric version of Tylers estimator (STyler) and provide an iterative fixed point algorithm to compute it. The classical results claim that at least n=p+1 sample points in general position are necessary to ensure the existence and uniqueness of Tylers estimator, where p is the ambient dimension. We show that the STyler requires significantly less samples. In some groups, even two samples are enough to guarantee its existence and uniqueness. In addition, in the case of elliptical populations, we provide high probability bounds on the error of the STyler. These, too, quantify the advantage of exploiting the symmetry structure. Finally, these theoretical results are supported by numerical simulations.

Performance Analysis of Tyler's Covariance Estimator

This paper analyzes the performance of Tylers M-estimator of the scatter matrix in elliptical populations. We focus on the non-asymptotic setting and derive estimation error bounds depending on the number of samples n and the dimension p. We show that under mild conditions the squared Frobenius norm of the error of the inverse estimator decays like p2/n with high probability.

Group symmetry and non-Gaussian covariance estimation

We consider robust covariance estimation with group symmetry constraints. Non-Gaussian covariance estimation, e.g., Tylers scatter estimator and Multivariate Generalized Gaussian distribution methods, usually involve non-convex minimization problems. Recently, it was shown that the underlying principle behind their success is an extended form of convexity over the geodesics in the manifold of positive definite matrices. A modern approach to improve estimation accuracy is to exploit prior knowledge via additional constraints, e.g., restricting the attention to specific classes of covariances which adhere to prior symmetry structures. In this paper, we prove that such group symmetry constraints are also geodesically convex and can therefore be incorporated into various non-Gaussian covariance estimators. Practical examples of such sets include: circulant, persymmetric and complex/quaternion proper structures. We provide a simple numerical technique for finding maximum likelihood estimates under such constraints, and demonstrate their performance advantage using synthetic experiments.

Invariance Theory for Adaptive Detection in Interference With Group Symmetric Covariance Matrix

This paper describes the adaptive detection of point-like targets in a Gaussian environment assuming a group symmetric structure for the interference covariance matrix. This special configuration enforces block-sparsity and permits splitting of the original observation space into lower-dimensional subspaces, each characterized by its own nuisance parameters. Hence, the Principle of Invariance is invoked to select decision rules enjoying some symmetry defined by a suitable group of transformations which leave the decision problem unaltered. All the invariant tests can be expressed as functions of a maximal invariant statistic whose derivation is among the key results of this paper. Finally, some common design criteria are applied to come up with adaptive architectures that share invariance, and their behavior is assessed to highlight the interplay between sample size and detection performance in comparison with conventional decision schemes.

Gaussian and robust Kronecker product covariance estimation

We study the Gaussian and robust covariance estimation, assuming the true covariance matrix to be a Kronecker product of two lower dimensional square matrices. In both settings we define the estimators as solutions to the constrained maximum likelihood programs. In the robust case, we consider Tylers estimator defined as the maximum likelihood estimator of a certain distribution on a sphere. We develop tight sufficient conditions for the existence and uniqueness of the estimates and show that in the Gaussian scenario with the unknown mean, p / q + q / p + 2 samples are almost surely enough to guarantee the existence and uniqueness, where p and q are the dimensions of the Kronecker product factors. In the robust case with the known mean, the corresponding sufficient number of samples is max p / q , q / p + 1 .

Joint covariance estimation with mutual linear structure

We consider the problem of joint estimation of structured covariance matrices. Assuming the structure is unknown, estimation is achieved using heterogeneous training sets. Namely, given groups of measurements coming from centered populations with different covariances, our aim is to determine the mutual structure of these covariance matrices and estimate them. Supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices, we develop a new efficient algorithm discovering the structure and using it to improve the estimation. Our technique is based on the application of principal component analysis in the matrix space. We also derive an upper performance bound of the proposed algorithm in the Gaussian scenario and compare it with the Cramér-Rao lower bound. Numerical simulations are presented to illustrate the performance benefits of the proposed method.

Joint Estimation of Inverse Covariance Matrices Lying in an Unknown Subspace

We consider the problem of joint estimation of inverse covariance matrices lying in an unknown subspace of the linear space of symmetric matrices. We perform the estimation using groups of measurements with different covariances. Assuming the inverse covariances span a low-dimensional subspace, our aim is to determine this subspace and to exploit this knowledge in order to improve the estimation. We develop a novel optimization algorithm discovering and exploiting the underlying low-dimensional subspace. We provide a computationally efficient algorithm and derive a tight upper performance bound. Numerical simulations on synthetic and real world data are presented to illustrate the performance benefits of the algorithm.

A channel matching based hybrid analog-digital strategy for massive multi-user MIMO downlink systems

In this paper we study the downlink of a hybrid analog-digital massive multi-user MIMO (MU-MIMO) system. An efficient hybrid strategy is proposed, where the analog beamforming matrices are determined using a channel matching criterion while the digital beamforming consists of pre-filters and post-filters. The digital post-filters are computed using traditional linear MU-MIMO strategies together with water-filling based power allocation using the effective channels. Simulation results show that the proposed hybrid analog-digital solutions achieve a good performance compared to their corresponding unconstrained digital solutions.

Joint Covariance Estimation With Mutual Linear Structure

We consider the problem of joint estimation of structured covariance matrices. Assuming the structure is unknown, estimation is achieved using heterogeneous training sets. Namely, given groups of measurements coming from centered populations with different covariances, our aim is to determine the mutual structure of these covariance matrices and estimate them. Supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices, we develop a new efficient algorithm discovering the structure and using it to improve the estimation. Our technique is based on the application of principal component analysis in the matrix space. We also derive an upper performance bound of the proposed algorithm in the Gaussian scenario and compare it with the Cramer–Rao lower bound. Numerical simulations are presented to illustrate the performance benefits of the proposed method.