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

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 ℝn generated by k-sparse vectors is bounded below by k lg (n/k) and above by 2k lg (n/k), plus some round-off terms. 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/k)) measurements, given only the signs of the measurement vector. This result holds for textitall probability measures on ℝn. It is further shown that random sign-flipping errors result only in an increase in the constant in the O(k lg (n/k)) estimate. Because constructing a consistent algorithm is not straight-forward, we present a heuristic based on the ℓ1-norm support vector machine, and illustrate that its computational performance is superior to a currently popular method.