An Approach to One-Bit Compressed Sensing Based on Probably Approximately Correct Learning Theory
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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 (n/k) and above by 2k (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 (n/k)) measurements, given only the signs of the measurement vector. This result holds for all probability measures on R^n. It is further shown that random sign-flipping errors result only in an increase in the constant in the O(k (n/k)) estimate. Because constructing a consistent algorithm is not straight-forward, we present a heuristic based on the ℓ_1-norm support vector machine, and illustrate that its computational performance is superior to a currently popular method.
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