### Abstract

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.

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 - Jan 1 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

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## Cite this

*Advances in Neural Information Processing Systems*,

*2015-January*, 1549-1557.