Abstract
Linear regression studies the problem of estimating a model parameter β∗ ∈ ℝ, from n observations {(yi,xi)}ni=1 from linear model yi = (xiβ∗) + ∈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.
Original language | English (US) |
---|---|
Pages (from-to) | 1549-1557 |
Number of pages | 9 |
Journal | Advances in Neural Information Processing Systems |
Volume | 2015-January |
State | Published - 2015 |
Event | 29th Annual Conference on Neural Information Processing Systems, NIPS 2015 - Montreal, Canada Duration: Dec 7 2015 → Dec 12 2015 |
ASJC Scopus subject areas
- Computer Networks and Communications
- Information Systems
- Signal Processing