Accurate Prediction of Phase Transitions in Compressed Sensing via a Connection to Minimax Denoising

Compressed sensing posits that, within limits, one can undersample a sparse signal and yet reconstruct it accurately. Knowing the precise limits to such undersampling is important both for theory and practice. We present a formula precisely delineating the allowable degree of of undersampling of generalized sparse objects. The formula applies to Approximate Message Passing (AMP) algorithms for compressed sensing, which are here generalized to employ denoising operators besides the traditional scalar shrinkers (soft thresholding, positive soft thresholding and capping). This paper gives several examples including scalar shrinkers not derivable from convex optimization -- the firm shrinkage nonlinearity and the minimax} nonlinearity -- and also nonscalar denoisers -- block thresholding (both block soft and block James-Stein), monotone regression, and total variation minimization. Let the variables \epsilon = k/N and \delta = n/N denote the generalized sparsity and undersampling fractions for sampling the k-generalized-sparse N-vector x_0 according to y=Ax_0. Here A is an n\times N measurement matrix whose entries are iid standard Gaussian. The formula states that the phase transition curve \delta = \delta(\epsilon) separating successful from unsuccessful reconstruction of x_0 by AMP is given by: \delta = M(\epsilon| Denoiser), where M(\epsilon| Denoiser) denotes the per-coordinate minimax mean squared error (MSE) of the specified, optimally-tuned denoiser in the directly observed problem y = x + z. In short, the phase transition of a noiseless undersampling problem is identical to the minimax MSE in a denoising problem.