Regularized Estimation of High-dimensional Covariance Matrices.
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Abstract
Many signal processing methods are fundamentally related to the
estimation of covariance matrices. In cases where there are a large
number of covariates the dimension of covariance matrices is much
larger than the number of available data samples. This is especially
true in applications where data acquisition is constrained by limited
resources such as time, energy, storage and bandwidth. This
dissertation attempts to develop necessary components for covariance
estimation in the high-dimensional setting. The dissertation makes
contributions in two main areas of covariance estimation: (1) high
dimensional shrinkage regularized covariance estimation and (2)
recursive online complexity regularized estimation with applications of
anomaly detection, graph tracking, and compressive sensing.
New shrinkage covariance estimation methods are proposed that
significantly outperform previous approaches in terms of mean squared
error. Two multivariate data scenarios are considered: (1)
independently Gaussian distributed data; and (2) heavy tailed
elliptically contoured data. For the former scenario we improve on
the Ledoit-Wolf (LW) shrinkage estimator using the principle of
Rao-Blackwell conditioning and iterative approximation of the
clairvoyant estimator. In the latter scenario, we apply a variance
normalizing transformation and propose an iterative robust LW
shrinkage estimator that is distribution-free within the elliptical
family. The proposed robustified estimator is implemented via fixed
point iterations with provable convergence and unique limit.
A recursive online covariance estimator is proposed for tracking
changes in an underlying time-varying graphical model. Covariance
estimation is decomposed into multiple decoupled adaptive regression
problems. A recursive recursive group lasso is derived using a
homotopy approach that generalizes online lasso methods to group
sparse system identification. By reducing the memory of the objective
function this leads to a group lasso regularized LMS that provably
dominates standard LMS. Finally, we introduce a state-of-the-art
sampling system, the Modulated Wideband Converter (MWC) which is based
on recently developed analog compressive sensing theory. By inferring
the block-sparse structures of the high-dimensional covariance matrix
from a set of random projections, the MWC is capable of achieving
sub-Nyquist sampling for multiband signals with arbitrary carrier
frequency over a wide bandwidth.Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/86396/1/yilun_1.pd