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Entropy-based covariance determinant estimation
© 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.An information-theoretic approach is described to estimate the determinant of the covariance matrix of a random vector sequence (a common task in a wide range of estimation and detection problems in signal processing for communications). The method is based on a prior entropy-based processing of the data using kernels and offers robustness against small-entropy contamination. The trade-off between optimality, accuracy and robustness is analyzed, along with the impact of the relative kernel bandwidth and data size.Peer ReviewedPostprint (published version
Law of Log Determinant of Sample Covariance Matrix and Optimal Estimation of Differential Entropy for High-Dimensional Gaussian Distributions
Differential entropy and log determinant of the covariance matrix of a
multivariate Gaussian distribution have many applications in coding,
communications, signal processing and statistical inference. In this paper we
consider in the high dimensional setting optimal estimation of the differential
entropy and the log-determinant of the covariance matrix. We first establish a
central limit theorem for the log determinant of the sample covariance matrix
in the high dimensional setting where the dimension can grow with the
sample size . An estimator of the differential entropy and the log
determinant is then considered. Optimal rate of convergence is obtained. It is
shown that in the case the estimator is asymptotically
sharp minimax. The ultra-high dimensional setting where is also
discussed.Comment: 19 page
Maximum Entropy Kernels for System Identification
A new nonparametric approach for system identification has been recently
proposed where the impulse response is modeled as the realization of a
zero-mean Gaussian process whose covariance (kernel) has to be estimated from
data. In this scheme, quality of the estimates crucially depends on the
parametrization of the covariance of the Gaussian process. A family of kernels
that have been shown to be particularly effective in the system identification
framework is the family of Diagonal/Correlated (DC) kernels. Maximum entropy
properties of a related family of kernels, the Tuned/Correlated (TC) kernels,
have been recently pointed out in the literature. In this paper we show that
maximum entropy properties indeed extend to the whole family of DC kernels. The
maximum entropy interpretation can be exploited in conjunction with results on
matrix completion problems in the graphical models literature to shed light on
the structure of the DC kernel. In particular, we prove that the DC kernel
admits a closed-form factorization, inverse and determinant. These results can
be exploited both to improve the numerical stability and to reduce the
computational complexity associated with the computation of the DC estimator.Comment: Extends results of 2014 IEEE MSC Conference Proceedings
(arXiv:1406.5706
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