Smoothed analysis in compressed sensing

Arbitrary matrices M ∈ R^m×n, randomly perturbed in an additive manner using a random matrix R ∈ R^m×n, are shown to asymptotically almost surely satisfy the so-called robust null space property. Whilst insisting on an asymptotically optimal order of magnitude for m required to attain unique reconstruction via ℓ₁-minimisation algorithms, our results track the level of arbitrariness allowed for the fixed seed matrix M as well as the degree of distributional irregularity allowed for the entries of the perturbing matrix R. Starting with sub-gaussian entries for R, our results culminate with these allowed to have substantially heavier tails than sub-exponential ones. Throughout this trajectory, two measures control the arbitrariness allowed for M; the first is ∥M∥∞ and the second is a localised notion of the Frobenius norm of M (which depends on the sparsity of the signal being reconstructed). A key tool driving our proofs is Mendelson’s small-ball method (Learning without concentration, J. ACM, Vol. 62, 2015).