Optimal Phase Transitions in Compressed Sensing

Compressed sensing deals with efficient recovery of analog signals from linear measurements. This paper presents a statistical study of compressed sensing by modeling the input signal as random processes. Three classes of encoders are considered, namely optimal nonlinear, optimal linear and random linear encoders. Focusing on optimal decoders, we investigate the fundamental tradeoff between measurement rate and reconstruction fidelity gauged by error probability and noise sensitivity in the absence and presence of measurement noise, respectively. Optimal phase transition thresholds are determined as a functional of the input distribution and compared to suboptimal thresholds achieved by various popular reconstruction algorithms. In particular, we show that Gaussian sensing matrices incur no penalty on the phase transition threshold with respect to optimal nonlinear encoding. Our results also provide a rigorous justification of previous results based on replica heuristics in the weak-noise regime.