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Similarity, Kernels and the Triangle Inequality

Jäkel, F. and Schölkopf, B. and Wichmann, F. A. :
Similarity, Kernels and the Triangle Inequality.
[Online-Edition: https://doi.org/10.1016/j.jmp.2008.03.001]
In: Journal of Mathematical Psychology, 52 (5) pp. 297-303.
[Article] , (2008)

Official URL: https://doi.org/10.1016/j.jmp.2008.03.001

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.

Item Type: Article
Erschienen: 2008
Creators: Jäkel, F. and Schölkopf, B. and Wichmann, F. A.
Title: Similarity, Kernels and the Triangle Inequality
Language: English
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.

Journal or Publication Title: Journal of Mathematical Psychology
Volume: 52
Number: 5
Divisions: 03 Department Human Sciences
03 Department Human Sciences > Institute for Psychology
03 Department Human Sciences > Institute for Psychology > Models of Higher Cognition
Date Deposited: 09 Jul 2018 09:13
DOI: 10.1016/j.jmp.2008.03.001
Official URL: https://doi.org/10.1016/j.jmp.2008.03.001
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