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