Similarity, kernels, and the triangle inequality |
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Authors: | Frank Jäkel Bernhard Schölkopf |
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Institution: | a Technische Universität Berlin, Fakultät IV, Sekr. 6-4, Franklinstr. 28/29, 10587 Berlin, Germany b Bernstein Center for Computational Neuroscience, Philippstr. 13, Haus 6, 10115 Berlin, Germany c Max Planck Institute for Biological Cybernetics, Spemannstr. 38, 72076 Tübingen, Germany |
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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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Keywords: | Kernel Similarity Triangle inequality Segmental additivity Multidimensional scaling Generalization |
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