The minimax noise sensitivity in compressed sensing

Consider the compressed sensing problem of estimating an unknown k-sparse n-vector from a set of m noisy linear equations. Recent work focused on the noise sensitivity of particular algorithms - the scaling of the reconstruction error with added noise. In this paper, we study the minimax noise sensitivity - the minimum is over all possible recovery algorithms and the maximum is over all vectors obeying a sparsity constraint. This fundamental quantity characterizes the difficulty of recovery when nothing is known about the vector other than the fact that it has at most k nonzero entries. Assuming random sensing matrices (i.i.d. Gaussian), we obtain non-asymptotic bounds which show that the minimax noise sensitivity is finite if m ≥ k + 3 and infinite if m ≤ k + 1. We also study the large system behavior where δ = m/n ∈ (0,1) denotes the undersampling fraction and k/n = ε ∈ (0,1) denotes the sparsity fraction. There is a phase transition separating successful and unsuccessful recovery: the minimax noise sensitivity is bounded for any δ > ε and is unbounded for any δ <; ε. One consequence of our results is that the Bayes optimal phase transitions of Wu and Verdu can be obtained uniformly over the class of all sparse vectors.