PASCAL - Pattern Analysis, Statistical Modelling and Computational Learning

SPARSE RECOVERY UNDER MATRIX UNCERTAINTY
M. Rosenbaum and A.B. Tsybakov
(2008) arxiv paper.

Abstract

We consider the model y = Xa + u; Z = X + v; where the random vector y and the random n x p matrix Z are observed, the n x p matrix X is unknown, v is an n x p random noise matrix, u is a noise independent of v, and a is a vector of unknown parameters to be estimated. The matrix uncertainty is in the fact that X is observed with additive error. For dimensions p that can be much larger than the sample size n we consider the estimation of sparse vectors a. Under the matrix uncertainty, the Lasso and Dantzig selector turn out to be extremely unstable in recovering the sparsity pattern (i.e., of the set of non-zero components of a), even if the noise level is very small. We suggest new estimators called the matrix uncertainty selectors (or shortly the MU-selectors) which are close to a in different norms and in the prediction risk if the Restricted Eigenvalue assumption on X is satisfied. We also show that under somewhat stronger assumptions these estimators recover correctly the sparsity pattern.

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Subjects:Computational, Information-Theoretic Learning with Statistics
Learning/Statistics & Optimisation
ID Code:6925
Deposited By:Alexandre Tsybakov
Deposited On:16 April 2010