Optimal linear estimation under unknown nonlinear transform

Abstract

Linear regression studies the problem of estimating a model parameter β∗∈\Rp, from n observations {(yi,xi)}ni=1 from linear model yi=⟨\xi,β∗⟩+ϵi. We consider a significant generalization in which the relationship between ⟨xi,β∗⟩ and yi 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 β∗ 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 yi and ⟨xi,β∗⟩. We also consider the high dimensional setting where β∗ 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 ⟨xi,β∗⟩ and yi, we establish minimax lower bounds that demonstrate the optimality of our estimators in both the classical and high dimensional regimes.