Sparse recovery of noisy data and Grothendieck inequality
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We present a detailed analysis of the unconstrained ℓ_1-method LASSO for sparse recovery of noisy data. The data is recovered by sensing its compressed output produced by randomly generated class of observing matrices satisfying a Restricted Isometry Property. We derive a new ℓ_1-error estimate which highlights the dependence on a certain compressiblity threshold: once the computed re-scaled residual crosses that threshold, the error is driven only by the (assumed small) noise and compressiblity. Here we identify the re-scaled residual as a key quantity which drives the error and we derive its sharp lower bound of order square-root of the size of the support of the computed solution. The essential bound is derived by Grothendieck inequality, in terms of integer quadratic form which involves the entry-wise signs of the computed solution.
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