One Scan 1-Bit Compressed Sensing

Based on $\alpha$-stable random projections with small $\alpha$, we develop a simple algorithm for compressed sensing (sparse signal recovery) by using only 1-bit (i.e., the sign) of the measurements. The method of $\alpha$-stable random projections has become popular in data stream computations. Using only 1-bit of the measurements results in substantial cost reduction in collection, storage, communication, and decoding for compressed sensing. The proposed algorithm is efficient in that the decoding procedure requires only one scan of the coordinates. For a $K$-sparse signal of length $N$, a conservative version of our algorithm requires $12.3K\log N$ measurements to recover the support and the signs of the signal. A more practical version needs fewer measurements, as validated by experiments. A closely-related issue is the estimation of $K$, i.e., the size of the support. It turns out that, the harmonic mean estimator developed in the prior work for $\alpha$-stable random projections already provides a very accurate estimate of $K$ for the task of sparse recovery, using merely (e.g.,) 5 or 10 measurements. Since this is an important practical problem, a separate technical note is provided to introduce very efficient estimators based on 1-bit or multi-bit measurements, for estimating $K$ as well as the scale parameter of the $\alpha$-stable distribution family for $0<\alpha\leq2$.