Distributed Decoding From Heterogeneous 1-Bit Compressive Measurements

We develop a communication-efficient distributed estimation for the 1-bit compressive sensing where unknown sparse signals are coded into binary measurements with noises and sign flips. We allow for distinctive sign-flipped probabilities and intensities of noises for measurements collected at different nodes, which raises a heterogeneity issue. We suggest a distributed algorithm through penalized least squares to recover sparse signals. This algorithm is computationally very efficient with only gradient information communicated. The resulting distributed estimate converges after a single iteration even when a lousy initial estimate is provided, and achieves a nearly oracle rate after a constant number of iterations. We prove that, under some mild conditions, with high probability, the distributed estimate approximates the underlying true sparse signal with precision δ after a finite number of iterations, as long as the total sample size N satisfies (s log p)/(δ2N)=O(1), where p is the dimension and s is the number of non-zero elements of the underlying true sparse signals. We establish statistical guarantee for support recovery. Extensive experiments are provided to illustrate the effectiveness of our proposed distributed algorithm. Supplementary materials to this article are available online.