A PAC learning approach to one-bit compressed sensing

In this paper, the problem of one-bit compressed sensing (OBCS) is formulated as a problem in probably approximately correct (PAC) learning theory. In particular, we study the set of half-spaces generated by sparse vectors, and derive explicit upper and lower bounds for the Vapnik- Chervonenkis (VC-) dimension. The upper bound implies that it is possible to achieve OBCS where the number of samples grows linearly with the sparsity dimension and logarithmically with the vector dimension, leaving aside issues of computational complexity. The lower bound implies that, for some choices of probability measures, at least this many samples are required.