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

In the problem of one-bit compressed sensing, the goal is to find a delta-close estimation of a k-sparse vector x in R^n given the signs of the entries of y = Phi x, where Phi is called the measurement matrix. For the one-bit compressed sensing problem, previous work [Plan, 2013][Gopi, 2013] achieved Theta (delta^{-2} k log(n/k)) and O~( 1/delta k log (n/k)) measurements, respectively, but the decoding time was Omega ( n k log (n/k)). In this paper, using tools and techniques developed in the context of two-stage group testing and streaming algorithms, we contribute towards the direction of sub-linear decoding time. We give a variety of schemes for the different versions of one-bit compressed sensing, such as the for-each and for-all versions, and for support recovery; all these have at most a log k overhead in the number of measurements and poly(k, log n) decoding time, which is an exponential improvement over previous work, in terms of the dependence on n

One-Bit ExpanderSketch for One-Bit Compressed Sensing

Is it possible to obliviously construct a set of hyperplanes H such that you can approximate a unit vector x when you are given the side on which the vector lies with respect to every h in H? In the sparse recovery literature, where x is approximately k-sparse, this problem is called one-bit compressed sensing and has received a fair amount of attention the last decade. In this paper we obtain the first scheme that achieves almost optimal measurements and sublinear decoding time for one-bit compressed sensing in the non-uniform case. For a large range of parameters, we improve the state of the art in both the number of measurements and the decoding time