On Fast Decoding of High-Dimensional Signals from One-Bit Measurements

In the problem of one-bit compressed sensing, the goal is to find an ǫ-close estimation of a ksparse vector x ∈ R given the signs of the entries of y = Φx, where Φ is called the measurement matrix. Similarly, in the problem of compressed sensing from phaseless measurements, the goal is to estimate a k-sparse vector x ∈ C given only the magnitudes of the measurements y = |Φx|. For the one-bit compressed sensing problem, previous work [PV13b, GNJN13] achieved Θ(ǫk log(n/k)) and Õ( ǫ k log(n/k)) measurements, respectively, in the for-all model, but the decoding time was Ω(nk log(n/k)). In this paper, using tools and techniques developed in the context of two-stage group testing, we contribute towards the direction of very fast decoding time. First, we give a simple scheme with O(k logn+ ǫk + logn log logk n log k ) measurements and decoding time poly(k, logn) in the for-each model. In addition, for the for-all version of the problem we give a scheme with O(k2 log(n/k) log logk n + ǫk log(n/k)) measurements, that allows decoding of the signal in poly(k, logn) time. For the phaseless measurements version in the for-all model, we give a recovery scheme with O(k2 log n k log logk n) measurements with the same running time. Furthermore, we show how to get an exponential improvement (in terms of n) in the decoding time of the support recovery algorithm in [GNJN13]. To the best of our knowledge, these are the first schemes that allow such a fast decoding time with a number of measurements that have such a small polynomial dependence on k.