Performance regions in compressed sensing from noisy measurements

In this paper, compressed sensing with noisy measurements is addressed. The theoretically optimal reconstruction error is studied by evaluating Tanaka's equation. The main contribution is to show that in several regions, which have different measurement rates and noise levels, the reconstruction error behaves differently. This paper also evaluates the performance of the belief propagation (BP) signal reconstruction method in the regions discovered. When the measurement rate and the noise level lie in a certain region, BP is suboptimal with respect to Tanaka's equation, and it may be possible to develop reconstruction algorithms with lower error in that region.