One-bit sensing, discrepancy and Stolarsky's principle

Abstract

A sign-linear one-bit map from the -dimensional sphere to the -dimensional Hamming cube is given by where . For , we estimate , the smallest integer so that there is a sign-linear map which has the -restricted isometric property, where we impose the normalized geodesic distance on and the Hamming metric on . Up to a polylogarithmic factor, , which has a dimensional correction in the power of . This is a question that arises from the one-bit sensing literature, and the method of proof follows from geometric discrepancy theory. We also obtain an analogue of the Stolarsky invariance principle for this situation, which implies that minimizing the -average of the embedding error is equivalent to minimizing the discrete energy , where is the normalized geodesic distance. Bibliography: 39 titles.