An Approach to One-Bit Compressed Sensing Based on Probably Approximately Correct Learning Theory

Abstract

In this paper, the problem of one-bit compressed sensing (OBCS) is formulated as a problem in probably approximately correct (PAC) learning. It is shown that the Vapnik-Chervonenkis (VC-) dimension of the set of half-spaces in $\R^n$ generated by $k$-sparse vectors is bounded below by $k ( \lfloor\lg (n/k) \rfloor +1 )$ and above by $\lfloor 2k \lg (en) \rfloor $. By coupling this estimate with well-established results in PAC learning theory, we show that a consistent algorithm can recover a $k$-sparse vector with $O(k \lg n)$ measurements, given only the signs of the measurement vector. This result holds for \textit{all} probability measures on $\R^n$. The theory is also applicable to the case of noisy labels, where the signs of the measurements are flipped with some unknown probability.