## Abstract

Consider the rank-1 spiked model: X = √νξu + Z, where ν is the spike intensity, u ∈ S^{k-1} is an unknown direction and ξ ∼ N(0, 1), Z ∼ N(0, I). Motivated by recent advances in analog-to-digital conversion, we study the problem of recovering u ∈ S^{k-1} from n i.i.d. modulo-reduced measurements Y = [X] mod ∆, focusing on the high-dimensional regime (k ≫ 1). We develop and analyze an algorithm that, for most directions u and ν = poly(k), estimates u to high accuracy using n = poly(k) measurements, provided that ∆ & √log k. Up to constants, our algorithm accurately estimates u at the smallest possible ∆ that allows (in an information-theoretic sense) to recover X from Y. A key step in our analysis involves estimating the probability that a line segment of length ≈ √ν in a random direction u passes near a point in the lattice ∆Z^{k}. Numerical experiments show that the developed algorithm performs well even in a non-asymptotic setting.

Original language | American English |
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Pages (from-to) | 1298-1320 |

Number of pages | 23 |

Journal | Proceedings of Machine Learning Research |

Volume | 151 |

State | Published - 2022 |

Event | 25th International Conference on Artificial Intelligence and Statistics, AISTATS 2022 - Virtual, Online, Spain Duration: 28 Mar 2022 → 30 Mar 2022 |

### Bibliographical note

Funding Information:This work was supported in part by ISF under Grant 1791/17 and in part by the GENESIS Consortium via the Israel Ministry of Economy and Industry. The work of Elad Romanov was supported in part by an Einstein-Kaye fellowship from the Hebrew University of Jerusalem.

Publisher Copyright:

Copyright © 2022 by the author(s)