1,887 research outputs found
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
Convex Banding of the Covariance Matrix
We introduce a new sparse estimator of the covariance matrix for
high-dimensional models in which the variables have a known ordering. Our
estimator, which is the solution to a convex optimization problem, is
equivalently expressed as an estimator which tapers the sample covariance
matrix by a Toeplitz, sparsely-banded, data-adaptive matrix. As a result of
this adaptivity, the convex banding estimator enjoys theoretical optimality
properties not attained by previous banding or tapered estimators. In
particular, our convex banding estimator is minimax rate adaptive in Frobenius
and operator norms, up to log factors, over commonly-studied classes of
covariance matrices, and over more general classes. Furthermore, it correctly
recovers the bandwidth when the true covariance is exactly banded. Our convex
formulation admits a simple and efficient algorithm. Empirical studies
demonstrate its practical effectiveness and illustrate that our exactly-banded
estimator works well even when the true covariance matrix is only close to a
banded matrix, confirming our theoretical results. Our method compares
favorably with all existing methods, in terms of accuracy and speed. We
illustrate the practical merits of the convex banding estimator by showing that
it can be used to improve the performance of discriminant analysis for
classifying sound recordings
Adaptive Thresholding for Sparse Covariance Matrix Estimation
In this paper we consider estimation of sparse covariance matrices and
propose a thresholding procedure which is adaptive to the variability of
individual entries. The estimators are fully data driven and enjoy excellent
performance both theoretically and numerically. It is shown that the estimators
adaptively achieve the optimal rate of convergence over a large class of sparse
covariance matrices under the spectral norm. In contrast, the commonly used
universal thresholding estimators are shown to be sub-optimal over the same
parameter spaces. Support recovery is also discussed. The adaptive thresholding
estimators are easy to implement. Numerical performance of the estimators is
studied using both simulated and real data. Simulation results show that the
adaptive thresholding estimators uniformly outperform the universal
thresholding estimators. The method is also illustrated in an analysis on a
dataset from a small round blue-cell tumors microarray experiment. A supplement
to this paper which contains additional technical proofs is available online.Comment: To appear in Journal of the American Statistical Associatio
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