  The 1-bit compressed sensing framework enables the recovery of a sparse
vector x from the sign information of each entry of its linear transformation.
Discarding the amplitude information can significantly reduce the amount of
data, which is highly beneficial in practical applications. In this paper, we
present a Bayesian approach to signal reconstruction for 1-bit compressed
sensing, and analyze its typical performance using statistical mechanics.
Utilizing the replica method, we show that the Bayesian approach enables better
reconstruction than the L1-norm minimization approach, asymptotically
saturating the performance obtained when the non-zero entries positions of the
signal are known. We also test a message passing algorithm for signal
reconstruction on the basis of belief propagation. The results of numerical
experiments are consistent with those of the theoretical analysis.
