Optimal linear estimation under unknown nonlinear transform

Abstract

Linear regression studies the problem of estimating a model parameter ^* R^p, from n observations \(y_i,x_i)\_i=1^n from linear model y_i = x_i,^* + _i. We consider a significant generalization in which the relationship between x_i,^* and y_i 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 y_i and x_i,^* . 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 x_i,^* and y_i, we establish minimax lower bounds that demonstrate the optimality of our estimators in both the classical and high dimensional regimes.

Tasks

Reproductions