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Sparse Recovery under Matrix Uncertainty | | Mathieu Rosenbaum
; Alexandre B. Tsybakov
; | | Date: |
15 Dec 2008 | | Abstract: | We consider the model y=X heta+e, Z=X+v, where the n-dimensional random
vector y and the n*p random matrix Z are observed, the n*p matrix X is unknown,
v is an n*p random noise matrix, e is a noise independent of v, and heta 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
heta. 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 heta), 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 heta 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. | | Source: | arXiv, 0812.2818 | | Services: | Forum | Review | PDF | Favorites | |
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