## Abstract

Locality Sensitive Hashing (LSH) is an effective method to index a set of points such that we can efficiently find the nearest neighbors of a query point. We extend this method to our novel Set-query LSH (SLSH), such that it can find the nearest neighbors of a set of points, given as a query. Let s(x, y) be the similarity between two points x and y. We define a similarity between a set Q and a point x by aggregating the similarities s(p, x) for all p ∈ Q. For example, we can take s(p, x) to be the angular similarity between p and x (i.e., 1 − z(x,p^{)} ), and aggregate by arithmetic π or geometric averaging, or taking the lowest similarity. We develop locality sensitive hash families and data structures for a large set of such arithmetic and geometric averaging similarities, and analyze their collision probabilities. We also establish an analogous framework and hash families for distance functions. Specifically, we give a structure for the euclidean distance aggregated by either averaging or taking the maximum. We leverage SLSH to solve a geometric extension of the approximate near neighbors problem. In this version, we consider a metric for which the unit ball is an ellipsoid and its orientation is specified with the query. An important application that motivates our work is group recommendation systems. Such a system embeds movies and users in the same feature space, and the task of recommending a movie for a group to watch together, translates to a set-query Q using an appropriate similarity.

Original language | English |
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Title of host publication | 17th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2020 |

Editors | Susanne Albers |

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

ISBN (Electronic) | 9783959771504 |

DOIs | |

State | Published - 1 Jun 2020 |

Event | 17th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2020 - Torshavn, Faroe Islands Duration: 22 Jun 2020 → 24 Jun 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 162 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 17th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2020 |
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Country/Territory | Faroe Islands |

City | Torshavn |

Period | 22/06/20 → 24/06/20 |

### Funding

Funders | Funder number |
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German-Israeli Foundation for Scientific Research and Development | 1367 |

Israel Science Foundation | 1595/19 |

## Keywords

- Distance functions
- Ellipsoid
- Group recommendations
- Locality sensitive hashing
- Nearest neighbors
- Similarity functions
- Similarity search