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## Abstract

Consider an unknown smooth function

*f*:[0,1]→R , and say we are given*n*noisy mod 1 samples of*f*, i.e.,*y*=(_{i}*f*(*x*)+_{i }*η*_{i }) mod 1 for*x*∈ [0,1] , where_{i }*η**denotes noise. Given the samples (*_{i}*x*,_{i}*y*)_{i}*n i*=1 , our goal is to recover smooth, robust estimates of the clean samples*f*(*x*) mod 1. We formulate a natural approach for solving this problem which works with angular embeddings of the noisy mod 1 samples over the unit complex circle, inspired by the angular synchronization framework. Our approach amounts to solving a quadratically constrained quadratic program (QCQP) which is NP-hard in its basic form, and therefore we consider its relaxation which is a trust region sub-problem and hence solvable efficiently. We demonstrate its robustness to noise via extensive numerical simulations on several synthetic examples, along with a detailed theoretical analysis. To the best of our knowledge, we provide the first algorithm for denoising mod 1 samples of a smooth function, which comes with robustness guarantees._{i}Original language | English |
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Title of host publication | Proceedings of Machine Learning Research |

Pages | 1868-1876 |

Number of pages | 25 |

Volume | 84 |

Publication status | Published - 9 Apr 2018 |

Event | The 21st International Conference on Artificial Intelligence and Statistics - Playa Blanca, Lanzarote, Canary Islands, Lanzarote, Spain Duration: 9 Apr 2018 → 11 Apr 2018 Conference number: 21 http://www.aistats.org/ |

### Conference

Conference | The 21st International Conference on Artificial Intelligence and Statistics |
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Abbreviated title | AISTATS 2018 |

Country/Territory | Spain |

City | Lanzarote |

Period | 9/04/18 → 11/04/18 |

Internet address |

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