Optimal linear estimation under unknown nonlinear transform

Linear regression studies the problem of estimating a model parameter β ^∗ ∈ ℝ, from n observations {(yi,xi)} ⁿ _i=1 from linear model yi = (x _i β ^∗ ) + ∈ _i . We consider a significant generalization in which the relationship between (x _i , 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 <x _i , 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 (x _i , β ^∗ } and y _i , we establish minimax lower bounds that demonstrate the optimality of our estimators in both the classical and high dimensional regimes.