Binary and Multi-Bit Coding for Stable Random Projections

The recent work [17] developed a 1-bit compressed sensing (CS) algorithm based on $α$-stable random projections. Although the work in [17] showed that the method is a strong competitor of other existing 1-bit algorithms, the procedure requires knowing $K$, the sparsity. Note that $K$ 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 $l_α$ norm of the signal, where $0<α\leq2$ 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.