## Abstract

We study density estimation for classes of shift-invariant distributions over R^{d}. A multidimensional distribution is “shift-invariant” if, roughly speaking, it is close in total variation distance to a small shift of it in any direction. Shift-invariance relaxes smoothness assumptions commonly used in non-parametric density estimation to allow jump discontinuities. The different classes of distributions that we consider correspond to different rates of tail decay. For each such class we give an efficient algorithm that learns any distribution in the class from independent samples with respect to total variation distance. As a special case of our general result, we show that d-dimensional shift-invariant distributions which satisfy an exponential tail bound can be learned to total variation distance error ε using Õ_{d}(1/ε^{d}^{+2}) examples and Õ_{d}(1/ε^{2}d^{+2}) time. This implies that, for constant d, multivariate log-concave distributions can be learned in Õ_{d}(1/ε^{2}d^{+2}) time using Õ_{d}(1/ε^{d}^{+2}) samples, answering a question of [29]. All of our results extend to a model of noise-tolerant density estimation using Huber’s contamination model, in which the target distribution to be learned is a (1 − ε, ε) mixture of some unknown distribution in the class with some other arbitrary and unknown distribution, and the learning algorithm must output a hypothesis distribution with total variation distance error O(ε) from the target distribution. We show that our general results are close to best possible by proving a simple Ω 1/ε^{d} information-theoretic lower bound on sample complexity even for learning bounded distributions that are shift-invariant.

Original language | English (US) |
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Title of host publication | 10th Innovations in Theoretical Computer Science, ITCS 2019 |

Editors | Avrim Blum |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770958 |

DOIs | |

State | Published - Jan 1 2019 |

Event | 10th Innovations in Theoretical Computer Science, ITCS 2019 - San Diego, United States Duration: Jan 10 2019 → Jan 12 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 124 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 10th Innovations in Theoretical Computer Science, ITCS 2019 |
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Country | United States |

City | San Diego |

Period | 1/10/19 → 1/12/19 |

## Keywords

- Density estimation
- Log-concave distributions
- Non-parametrics
- Unsupervised learning

## ASJC Scopus subject areas

- Software