ACCURATE PREDICTION OF PHASE TRANSITIONS IN COMPRESSED SENSING VIA A CONNECTION TO MINIMAX DENOISING By

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 penalization – 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 ε = k/N and δ = n/N denote the generalized sparsity and undersampling fractions for sampling the k-generalized-sparse N -vector x0 according to y = Ax0. Here A is an n × N measurement matrix whose entries are iid standard Gaussian. The formula states that the phase transition curve δ = δ(ε) separating successful from unsuccessful reconstruction of x0 by AMP is given by: δ =M(ε|Denoiser), where M(ε|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. We derive this formula from State Evolution and we present numerical results validating the formula in a wide range of settings. The above formula generates numerous insights, including: (a) in the scalar sparsity case, that `1 minimization is very close to optimal among all possible sparsity seeking algorithms, including those deriving from nonconvex penalization; (b) in the block sparsity case, block soft thresholding is dramatically outperformed by James-Stein block thresholding for large blocksizes; block James-Stein recovers block sparse signals for any δ > ε, provided the blocklength is above a finite value B∗(δ, ε). (c) in the nonseparable cases of monotone regression and TV denoising, the sparsity-undersampling phase transition obtained by AMP tailored to the generalized sparsity is definitely better than than the phase transition achievable using merely `1 minimization of coefficients in the Haar basis. ∗Department of Statistics, Stanford University †Department of Electrical Engineering, Stanford University