Sparse recovery of noisy data using the Lasso method

We present a detailed analysis of the unconstrained $ell_1$-method Lasso method 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 $ell_1$-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.