### Abstract

We investigate a distance metric, previously defined for the measurement of structured data, in the more general context of vector spaces. The metric has a basis in information theory and assesses the distance between two vectors in terms of their relative information content. The resulting metric gives an outcome based on the dimensional correlation, rather than magnitude, of the input vectors, in a manner similar to Cosine Distance. In this paper the metric is defined, and assessed, in comparison with Cosine Distance, for its major properties: semantics, properties for use within similarity search, and evaluation efficiency.

We find that it is fairly well correlated with Cosine Distance in dense spaces, but its semantics are in some cases preferable. In a sparse space, it significantly outperforms Cosine Distance over TREC data and queries, the only large collection for which we have a human-ratified ground truth. This result is backed up by another experiment over movielens data. In dense Cartesian spaces it has better properties for use with similarity indices than either Cosine or Euclidean Distance. In its definitional form it is very expensive to evaluate for high-dimensional sparse vectors; to counter this, we show an algebraic rewrite which allows its evaluation to be performed more efficiently.

Overall, when a multivariate correlation metric is required over positive vectors, SED seems to be a better choice than Cosine Distance in many circumstances.

We find that it is fairly well correlated with Cosine Distance in dense spaces, but its semantics are in some cases preferable. In a sparse space, it significantly outperforms Cosine Distance over TREC data and queries, the only large collection for which we have a human-ratified ground truth. This result is backed up by another experiment over movielens data. In dense Cartesian spaces it has better properties for use with similarity indices than either Cosine or Euclidean Distance. In its definitional form it is very expensive to evaluate for high-dimensional sparse vectors; to counter this, we show an algebraic rewrite which allows its evaluation to be performed more efficiently.

Overall, when a multivariate correlation metric is required over positive vectors, SED seems to be a better choice than Cosine Distance in many circumstances.

Original language | English |
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Title of host publication | Similarity search and applications |

Subtitle of host publication | 5th international conference, SISAP 2012 proceedings |

Editors | Gonzalo Nararro, Vladimir Pestov |

Place of Publication | Berlin |

Publisher | Springer-Verlag |

Pages | 209-225 |

Number of pages | 17 |

ISBN (Print) | 9783642321528 |

DOIs | |

Publication status | Published - 2012 |

Event | 5th International Conference on Similarity Search and Applications (SISAP) - Toronto, Canada Duration: 9 Aug 2012 → 10 Aug 2012 |

### Publication series

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

Volume | 7404 |

ISSN (Print) | 0302-9743 |

### Conference

Conference | 5th International Conference on Similarity Search and Applications (SISAP) |
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Country | Canada |

City | Toronto |

Period | 9/08/12 → 10/08/12 |

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### Keywords

- correlation distance
- vector spaces
- multivariate
- distance metric
- similarity search
- cosine distance
- multivariate correlation

### Cite this

Connor, R., & Moss, R. G. (2012). A multivariate correlation distance for vector spaces. In G. Nararro, & V. Pestov (Eds.),

*Similarity search and applications: 5th international conference, SISAP 2012 proceedings*(pp. 209-225). (Lecture Notes in Computer Science; Vol. 7404). Berlin: Springer-Verlag. https://doi.org/10.1007/978-3-642-32153-5_15