29,964 research outputs found
Adaptive covariance matrix estimation through block thresholding
Estimation of large covariance matrices has drawn considerable recent
attention, and the theoretical focus so far has mainly been on developing a
minimax theory over a fixed parameter space. In this paper, we consider
adaptive covariance matrix estimation where the goal is to construct a single
procedure which is minimax rate optimal simultaneously over each parameter
space in a large collection. A fully data-driven block thresholding estimator
is proposed. The estimator is constructed by carefully dividing the sample
covariance matrix into blocks and then simultaneously estimating the entries in
a block by thresholding. The estimator is shown to be optimally rate adaptive
over a wide range of bandable covariance matrices. A simulation study is
carried out and shows that the block thresholding estimator performs well
numerically. Some of the technical tools developed in this paper can also be of
independent interest.Comment: Published in at http://dx.doi.org/10.1214/12-AOS999 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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
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