Optimal linear estimation under unknown nonlinear transform
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Linear regression studies the problem of estimating a model parameter β^* ∈R^p, from n observations {(y_i,x_i)}_i=1^n from linear model y_i = 〈x_i,β^* 〉 + ϵ_i. We consider a significant generalization in which the relationship between 〈x_i,β^* 〉 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 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 y_i 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.
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