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Optimal linear estimation under unknown nonlinear transform

Author(s): Yi, X; Wang, Z; Caramanis, C; Liu, H

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Abstract: Linear regression studies the problem of estimating a model parameter β∗∈\Rp, from n observations {(yi,xi)}ni=1 from linear model yi=⟨\xi,β∗⟩+ϵi. 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.
Publication Date: 2015
Citation: Yi, Xinyang, Zhaoran Wang, Constantine Caramanis, and Han Liu. "Optimal linear estimation under unknown nonlinear transform." In Advances in neural information processing systems 28, pp. 1549-1557. 2015.
ISSN: 1049-5258
Pages: 1549 - 1557
Type of Material: Conference Article
Journal/Proceeding Title: Advances in Neural Information Processing Systems
Version: Author's manuscript

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