Compressed sensing with corrupted observations

We proposed a weighted l minimization: min , ‖x‖ + λ‖f‖ s.t.Ax+ f= b to recover a sparse vector x and the corrupted noise vector f from a linear measurement b = Ax + f when the sensing matrix A is an m × n row i.i.d subgaussian matrix. Our first result shows that the recovery is possible when the fraction of corrupted noise is smaller than a positive constant, provided that ‖x‖ ≤ O(n/ln (n/‖x ∗‖ ), which is also the asymptotically optimal bound. While our second result shows that the recovery is still possible when the fraction of corruption noise grows arbitrary close to 1, as long as m ≥ O‖x‖ ln(‖x ∗‖ ), which is asymptotically better than the bound m ≥ O‖x‖ ln(n)ln (m ) achieved by a recent literature [1] by a ln (n) factor.