An approach to one-bit compressed sensing based on probably approximately correct learning theory

This paper builds upon earlier work of the authors in formulating the one-bit compressed sensing (OBCS) problem as a problem in probably approximately correct (PAC) learning theory. It is shown that the solution to the OBCS problem consists of two parts. The first part is to determine the statistical complexity of OBCS by determining the Vapnik-Chervonenkis (VC-) dimension of the set of half-spaces generated by sparse vectors. The second is to determine the algorithmic complexity of the problem by developing a “consistent” algorithm. In this paper, we generalize the earlier results of the authors by deriving both upper and lower bounds on the VC-dimension of half-spaces generated by sparse vectors, even when the separating hyperplane need not pass through the origin. As with earlier bounds, these bounds grow linearly with respect to with the sparsity dimension and logarithmically with the vector dimension.