Optimal Linear Estimation under Unknown Nonlinear Transform

Linear regression studies the problem of estimating a model parameter β* ∈ℝ p , from n observations [Formula: see text] from linear model yi = 〈xi , β*〉 + ε i . 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.