### Abstract

In a metric space, triangle inequality implies that, for any three objects, a triangle with edge lengths corresponding to their pair- wise distances can be formed. The n-point property is a generalisation of this where, for any (n + 1) objects in the space, there exists an n- dimensional simplex whose edge lengths correspond to the distances among the objects. In general, metric spaces do not have this prop- erty; however in 1953, Blumenthal showed that any semi-metric space which is isometrically embeddable in a Hilbert space also has the n-point property.

We have previously called such spaces supermetric spaces, and have shown that many metric spaces are also supermetric, including Euclidean, Cosine, Jensen-Shannon and Triangular spaces of any dimension.

Here we show how such simplexes can be constructed from only their edge lengths, and we show how the geometry of the simplexes can be used to determine lower and upper bounds on unknown distances within the original space. By increasing the number of dimensions, these bounds converge to the true distance.

Finally we show that for any Hilbert-embeddable space, it is possible to construct Euclidean spaces of arbitrary dimensions, from which these lower and upper bounds of the original space can be determined. These spaces may be much cheaper to query than the original. For similarity search, the engineering tradeoffs are good: we show significant reductions in data size and metric cost with little loss of accuracy, leading to a significant overall improvement in exact search performance.

We have previously called such spaces supermetric spaces, and have shown that many metric spaces are also supermetric, including Euclidean, Cosine, Jensen-Shannon and Triangular spaces of any dimension.

Here we show how such simplexes can be constructed from only their edge lengths, and we show how the geometry of the simplexes can be used to determine lower and upper bounds on unknown distances within the original space. By increasing the number of dimensions, these bounds converge to the true distance.

Finally we show that for any Hilbert-embeddable space, it is possible to construct Euclidean spaces of arbitrary dimensions, from which these lower and upper bounds of the original space can be determined. These spaces may be much cheaper to query than the original. For similarity search, the engineering tradeoffs are good: we show significant reductions in data size and metric cost with little loss of accuracy, leading to a significant overall improvement in exact search performance.

Original language | English |
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Title of host publication | Similarity Search and Applications |

Subtitle of host publication | 10th International Conference, SISAP 2017, Munich, Germany, October 4-6, 2017, Proceedings |

Editors | Christian Beecks, Peer Kröger, Thomas Seidl |

Place of Publication | Cham |

Publisher | Springer |

Pages | 96-109 |

Number of pages | 14 |

Volume | 10609 |

ISBN (Print) | 9783319684734 |

DOIs | |

Publication status | Published - 4 Oct 2017 |

Event | SISAP 2017: 10th International Conference on Similarity Search and Applications - Munich, Germany Duration: 4 Oct 2017 → 6 Oct 2017 |

### Publication series

Name | Lecture Notes in Computer Science |
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Publisher | Springer Verlag |

Volume | 10609 |

ISSN (Print) | 0302-9743 |

### Conference

Conference | SISAP 2017 |
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Country | Germany |

City | Munich |

Period | 4/10/17 → 6/10/17 |

### Keywords

- supermetric space
- metric search
- metric embedding
- dimensionality reduction
- distance geometry

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## Cite this

Connor, R., Vadicamo, L., & Rabitti, F. (2017). High-dimensional simplexes for metric search. In C. Beecks, P. Kröger, & T. Seidl (Eds.),

*Similarity Search and Applications: 10th International Conference, SISAP 2017, Munich, Germany, October 4-6, 2017, Proceedings*(Vol. 10609, pp. 96-109). (Lecture Notes in Computer Science; Vol. 10609). Springer. https://doi.org/10.1007/978-3-319-68474-1_7