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  Similarity, Kernels, and the Triangle Inequality

Jäkel, F., Schölkopf, B., & Wichmann, F. (2008). Similarity, Kernels, and the Triangle Inequality. Journal of Mathematical Psychology, 52(2), 297-303. doi:10.1016/j.jmp.2008.03.001.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0013-C71B-F Version Permalink: http://hdl.handle.net/21.11116/0000-0003-2D0F-C
Genre: Journal Article

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Jäkel, F, Author              
Schölkopf, B1, 2, Author              
Wichmann, FA, Author              
Affiliations:
1Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society, ou_1497795              
2Max Planck Institute for Biological Cybernetics, Max Planck Society, Spemannstrasse 38, 72076 Tübingen, DE, ou_1497794              

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 Abstract: Similarity is used as an explanatory construct throughout psychology and multidimensional scaling (MDS) is the most popular way to assess similarity. In MDS, similarity is intimately connected to the idea of a geometric representation of stimuli in a perceptual space. Whilst connecting similarity and closeness of stimuli in a geometric representation may be intuitively plausible, Tversky and Gati [Tversky, A., Gati, I. (1982). Similarity, separability, and the triangle inequality. Psychological Review, 89(2), 123–154] have reported data which are inconsistent with the usual geometric representations that are based on segmental additivity. We show that similarity measures based on Shepard’s universal law of generalization [Shepard, R. N. (1987). Toward a universal law of generalization for psychologica science. Science, 237(4820), 1317–1323] lead to an inner product representation in a reproducing kernel Hilbert space. In such a space stimuli are represented by their similarity to all other stimuli. This representation, based on Shepard’s law, has a natural metric that does not have additive segments whilst still retaining the intuitive notion of connecting similarity and distance between stimuli. Furthermore, this representation has the psychologically appealing property that the distance between stimuli is bounded.

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 Dates: 2008-09
 Publication Status: Published in print
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 Rev. Method: -
 Identifiers: DOI: 10.1016/j.jmp.2008.03.001
BibTex Citekey: 4785
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Title: Journal of Mathematical Psychology
Source Genre: Journal
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Publ. Info: Orlando, Fla. : Academic Press
Pages: - Volume / Issue: 52 (2) Sequence Number: - Start / End Page: 297 - 303 Identifier: ISSN: 0022-2496
CoNE: https://pure.mpg.de/cone/journals/resource/954922646040