Abstract

 Linear regression studies the problem of estimating a model parameter , from n observations  from linear model     . We consider a significant generalization in which the relationship between and is noisy, quantized to a single bit, potentially nonlinear, noninvertible, as well as unknown. This model is known as the single-index model in statistics, and, among other things, it represents a significant generalization of one-bit compressed sensing. We propose a novel spectral-based estimation procedure and show that we can recover in settings (i.e., classes of link function f ) where previous algorithms fail. In general, our algorithm requires only very mild restrictions on the (unknown) functional relationship between . We also consider the high dimensional setting where is sparse, and introduce a two-stage nonconvex framework that addresses estimation challenges in high dimensional regimes where p  n. For a broad class of link functions between , we establish minimax lower bounds that demonstrate the optimality of our estimators in both the classical and high dimensional regimes.