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

Abstract

In the problem of one-bit compressed sensing, the goal is to find a $\delta$-close estimation of a $k$-sparse vector $x \in \mathbb{R}^n$ 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 \cite{Plan-robust,support} achieved $\Theta (\delta^{-2} k \log(n/k))$ and $\tilde{ \Oh} ( \frac{1}{ \delta } k \log (n/k))$ 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 very fast 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 version, support recovery; all these have $poly(k, \log n)$ decoding time, which is an exponential improvement over previous work, in terms of the dependence of $n$.