30,817 research outputs found
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 Tyler's 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 Tyler's estimator and its
projection onto the structure set.Comment: arXiv admin note: text overlap with arXiv:1311.059
Covariance Estimation in Elliptical Models with Convex Structure
We address structured covariance estimation in Elliptical distribution. We
assume it is a priori known that the covariance belongs to a given convex set,
e.g., the set of Toeplitz or banded matrices. We consider the General Method of
Moments (GMM) optimization subject to these convex constraints. Unfortunately,
GMM is still non-convex due to objective. Instead, we propose COCA - 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 as Tyler's estimate and its
projection onto a convex set
User-Friendly Covariance Estimation for Heavy-Tailed Distributions
We offer a survey of recent results on covariance estimation for heavy-tailed
distributions. By unifying ideas scattered in the literature, we propose
user-friendly methods that facilitate practical implementation. Specifically,
we introduce element-wise and spectrum-wise truncation operators, as well as
their -estimator counterparts, to robustify the sample covariance matrix.
Different from the classical notion of robustness that is characterized by the
breakdown property, we focus on the tail robustness which is evidenced by the
connection between nonasymptotic deviation and confidence level. The key
observation is that the estimators needs to adapt to the sample size,
dimensionality of the data and the noise level to achieve optimal tradeoff
between bias and robustness. Furthermore, to facilitate their practical use, we
propose data-driven procedures that automatically calibrate the tuning
parameters. We demonstrate their applications to a series of structured models
in high dimensions, including the bandable and low-rank covariance matrices and
sparse precision matrices. Numerical studies lend strong support to the
proposed methods.Comment: 56 pages, 2 figure
A new family of high-resolution multivariate spectral estimators
In this paper, we extend the Beta divergence family to multivariate power
spectral densities. Similarly to the scalar case, we show that it smoothly
connects the multivariate Kullback-Leibler divergence with the multivariate
Itakura-Saito distance. We successively study a spectrum approximation problem,
based on the Beta divergence family, which is related to a multivariate
extension of the THREE spectral estimation technique. It is then possible to
characterize a family of solutions to the problem. An upper bound on the
complexity of these solutions will also be provided. Simulations suggest that
the most suitable solution of this family depends on the specific features
required from the estimation problem
A Geometric Approach to Covariance Matrix Estimation and its Applications to Radar Problems
A new class of disturbance covariance matrix estimators for radar signal
processing applications is introduced following a geometric paradigm. Each
estimator is associated with a given unitary invariant norm and performs the
sample covariance matrix projection into a specific set of structured
covariance matrices. Regardless of the considered norm, an efficient solution
technique to handle the resulting constrained optimization problem is
developed. Specifically, it is shown that the new family of distribution-free
estimators shares a shrinkagetype form; besides, the eigenvalues estimate just
requires the solution of a one-dimensional convex problem whose objective
function depends on the considered unitary norm. For the two most common norm
instances, i.e., Frobenius and spectral, very efficient algorithms are
developed to solve the aforementioned one-dimensional optimization leading to
almost closed form covariance estimates. At the analysis stage, the performance
of the new estimators is assessed in terms of achievable Signal to Interference
plus Noise Ratio (SINR) both for a spatial and a Doppler processing assuming
different data statistical characterizations. The results show that interesting
SINR improvements with respect to some counterparts available in the open
literature can be achieved especially in training starved regimes.Comment: submitted for journal publicatio
Group Symmetry and non-Gaussian Covariance Estimation
We consider robust covariance estimation with group symmetry constraints.
Non-Gaussian covariance estimation, e.g., Tyler 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
Relaxed concentrated MLE for robust calibration of radio interferometers
In this paper, we investigate the calibration of radio interferometers in
which Jones matrices are considered to model the interaction between the
incident electromagnetic field and the antennas of each station. Specifically,
perturbation effects are introduced along the signal path, leading to the
conversion of the plane wave into an electric voltage by the receptor. In order
to design a robust estimator, the noise is assumed to follow a spherically
invariant random process (SIRP). The derived algorithm is based on an iterative
relaxed concentrated maximum likelihood estimator (MLE), for which closed-form
expressions are obtained for most of the unknown parameters
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