Optimal linear estimation under unknown nonlinear transform

摘要: Linear regression studies the problem of estimating a model parameter β* ∈ ℝp, from n observations {(yi, xi)}ni=1 linear yi = 〈xi, β*〉 + ∊i. We consider significant generalization in which relationship between and is noisy, quantized to single bit, potentially nonlinear, noninvertible, as well unknown. This known single-index statistics, and, among other things, it represents one-bit compressed sensing. propose novel spectral-based estimation procedure show that we can recover settings (i.e., classes link function f) where previous algorithms fail. In general, our algorithm requires only very mild restrictions on (unknown) functional β*〉. also high dimensional setting sparse, introduce two-stage nonconvex framework addresses challenges regimes p ≫ n. For broad class functions yi, establish minimax lower bounds demonstrate optimality estimators both classical regimes.