Binary and Multi-Bit Coding for Stable Random Projections

The recent work [17] developed a 1-bit compressed sensing (CS) algorithm based on alpha-stable random projections. Although it was shown in [17] that the method is a strong competitor to other existing 1-bit CS algorithms, the procedure requires knowing K, the sparsity, which is the l(0) norm of the signal. Other existing 1-bit CS algorithms require the l(2) norm of the signal. In this paper, we develop an estimation procedure for the l(alpha) norm of the signal, where 0 < alpha <= 2 from binary or multi-bit measurements. We demonstrate that using a simple closed-form estimator with merely 1-bit information does not result in a significant loss of accuracy if the parameter is chosen appropriately. Theoretical tail bounds are also provided. Using 2 or more bits per measurement reduces the variance and importantly, stabilizes the estimate so that the variance is not too sensitive to chosen parameters.