Hamming Compressed Sensing

Compressed sensing (CS) and 1-bit CS cannot directly recover quantized signals and require time consuming recovery. In this paper, we introduce Hamming compressed sensing (HCS) that directly recovers a k-bit quantized signal of dimensional n from its 1-bit measurements via invoking n times of Kullback-Leibler divergence based nearest neighbor search. Compared with CS and 1-bit CS, HCS allows the signal to be dense, takes considerably less (linear) recovery time and requires substantially less measurements (O(log n)). Moreover, HCS recovery can accelerate the subsequent 1-bit CS dequantizer. We study a quantized recovery error bound of HCS for general signals and “HCS+dequantizer” recovery error bound for sparse signals. Extensive numerical simulations verify the appealing accuracy, robustness, efficiency and consistency of HCS. Index Terms Compressed sensing, 1-bit compressed sensing, HCS quantizer, quantized recovery, nearest neighbor search, dequantizer.