Information theoretic performance bounds for noisy compressive sensing

Compressive sensing provides a new approach to data acquisition and storage. In this paper, we derive some information theory bounds on the performance of noisy compressive sensing to calculate the data rate with particular distortion, which has significant meaning in data storage technique. We analyze the rate distortion performance of noisy compressive sensing under Mean Squared distortion and Hamming distortion, and give more accurate results. Besides, mathematical lower bounds of rate distortion function and theoretical minimal useful bit rates are provided for these two distortion for the first time. We also give a theoretical upper bound of the Mean Squared distortion of compressive sensing process. The relationships of bit rate per dimension R(D)/N and M, N, and M/N are given and plotted in this paper, and both theoretical analysis and numerical results show that compressive sensing uses less number of bits to represent the same information compared to conventional information acquisition and reconstruction techniques.