Abstract
There are many contexts where the definition of similarity in multivariate space requires to be based on the correlation, rather than absolute value, of the variables. Examples include classic IR measurements such as TDF/IF and BM25, client similarity measures based on collaborative filtering, feature analysis of chemical molecules, and biodiversity contexts.
In such cases, it is almost standard for Cosine similarity to be used. More recently, Jensen-Shannon divergence has appeared in a proper metric form, and a related metric Structural Entropic Distance (SED) has been investigated. A fourth metric, based on a little-known divergence function named as Triangular Divergence, is also assessed here.
For these metrics, we study their properties in the context of similarity and metric search. We compare and contrast their semantics and performance. Our conclusion is that, despite Cosine Distance being an almost automatic choice in this context, Triangular Distance is most likely to be the best choice in terms of a compromise between semantics and performance.
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Notes
- 1.
Some functions are formally undefined in the presence of zero values, requiring either \(0\log 0\) or 0/0. In each case, there is in fact a good mathematical argument for treating these terms as 0 rather than undefined.
- 2.
See [4] for an explanation of this constant.
- 3.
In fact Shannon’s entropy raised to the power of the logarithm base, see [4] for details.
- 4.
118 dimensions.
- 5.
References
Connor, R., Cardillo, F.A., Vadicamo, L., Rabitti, F.: Hilbert Exclusion: Improved Metric Search Through Finite Isometric Embeddings. ArXiv e-prints, accepted for publication ACM TOIS, April 2016
Connor, R., Cardillo, F.A., Vadicamo, L., Rabitti, F.: Supermetric Search with the Four-Point Property. Accepted for publication SISAP, Tokyo, Japan, October 2016
Connor, R., Moss, R.: A multivariate correlation distance for vector spaces. In: Navarro, G., Pestov, V. (eds.) SISAP 2012. LNCS, vol. 7404, pp. 209–225. Springer, Heidelberg (2012)
Connor, R., Simeoni, F., Iakovos, M., Moss, R.: A bounded distance metric for comparing tree structure. Inf. Syst. 36(4), 748–764 (2011)
Connor, R., Moss, R., Harvey, M.: A new probabilistic ranking model. In: Proceedings of the 2013 Conference on the Theory of Information Retrieval, ICTIR 2013, p. 23: 109–23: 112, NY, USA (2013). http://doi.acm.org/10.1145/2499178.2499185
Endres, D., Schindelin, J.: A new metric for probability distributions. IEEE Trans. Inf. Theor. 49(7), 1858–1860 (2003)
Fuglede, B., Topsoe, F.: Jensen-Shannon divergence and Hilbert space embedding. In: Proceedings of International Symposium on Information Theory, ISIT 2004, p. 31 (2004)
Jones, K.S., Walker, S., Robertson, S.E.: A probabilistic model of information retrieval: development and comparative experiments: part 2. Inf. Process. Manag. 36(6), 809–840 (2000)
Lin, J.: Divergence measures based on the Shannon entropy. IEEE Trans. Inf. Theor. 37(1), 145–151 (1991)
Österreicher, F., Vajda, I.: A new class of metric divergences on probability spaces and and its statistical applications. Ann. Inst. Stat. Math. 55, 639–653 (2003)
Singhal, A.: Modern information retrieval: a brief overview. IEEE Data Eng. Bull. 24(4), 35–43 (2001)
Topsoe, F.: Some inequalities for information divergence and related measures of discrimination. IEEE Trans. Inf. Theor. 46(4), 1602–1609 (2000)
Topsøe, F.: Jenson-Shannon divergence and norm-based measures of discrimination and variation. Preprint math.ku.dk (2003)
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Connor, R. (2016). A Tale of Four Metrics. In: Amsaleg, L., Houle, M., Schubert, E. (eds) Similarity Search and Applications. SISAP 2016. Lecture Notes in Computer Science(), vol 9939. Springer, Cham. https://doi.org/10.1007/978-3-319-46759-7_16
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DOI: https://doi.org/10.1007/978-3-319-46759-7_16
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