## Abstract

In many situations, sample data is obtained from a noisy or imperfect source. In order to address such corruptions, this paper introduces the concept of a sampling corrector. Such algorithms use structure that the distribution is purported to have, in order to allow one to make "on-The-fly" corrections to samples drawn from probability distributions. These algorithms then act as filters between the noisy data and the end user. We show connections between sampling correctors, distribution learning algorithms, and distribution property testing algorithms. We show that these connections can be utilized to expand the applicability of known distribution learning and property testing algorithms as well as to achieve improved algorithms for those tasks. As a first step, we show how to design sampling correctors using proper learning algorithms. We then focus on the question of whether algorithms for sampling correctors can be more efficient in terms of sample complexity than learning algorithms for the analogous families of distributions. When correcting monotonicity, we show that this is indeed the case when also granted query access to the cumulative distribution function. We also obtain sampling correctors for monotonicity without this stronger type of access, provided that the distribution be originally very close to monotone (namely, at a distance O(1/ log^{2} n)). In addition to that, we consider a restricted error model that aims at capturing "missing data" corruptions. In this model, we show that distributions that are close to monotone have sampling correctors that are significantly more efficient than achievable by the learning approach. We then consider the question of whether an additional source of independent random bits is required by sampling correctors to implement the correction process. We show that for correcting close-Touniform distributions and close-To-monotone distributions, no additional source of random bits is required, as the samples from the input source itself can be used to produce this randomness.

Original language | English |
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Title of host publication | ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science |

Publisher | Association for Computing Machinery, Inc |

Pages | 93-102 |

Number of pages | 10 |

ISBN (Electronic) | 9781450340571 |

DOIs | |

State | Published - 14 Jan 2016 |

Event | 7th ACM Conference on Innovations in Theoretical Computer Science, ITCS 2016 - Cambridge, United States Duration: 14 Jan 2016 → 16 Jan 2016 |

### Publication series

Name | ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science |
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### Conference

Conference | 7th ACM Conference on Innovations in Theoretical Computer Science, ITCS 2016 |
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Country/Territory | United States |

City | Cambridge |

Period | 14/01/16 → 16/01/16 |

## Keywords

- Learning
- Probability distributions
- Property testing
- Randomized algorithms