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Empirical Inference Article 2008

Similarity, Kernels, and the Triangle Inequality

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Empirical Inference
Bernhard Schölkopf
  • Director
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Empirical Inference
Frank Jäkel
Thumb ticker sm thumb felix a. wichmann
Empirical Inference
Felix A. Wichmann
  • Research Scientist

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.

Author(s): Jäkel, F. and Schölkopf, B. and Wichmann, FA.
Journal: Journal of Mathematical Psychology
Volume: 52
Number (issue): 5
Pages: 297-303
Year: 2008
Month: September
Day: 0
Bibtex Type: Article (article)
DOI: 10.1016/j.jmp.2008.03.001
Digital: 0
Electronic Archiving: grant_archive
Language: en
Organization: Max-Planck-Gesellschaft
School: Biologische Kybernetik
Links:

BibTex 

@article{4785,
  title = {Similarity, Kernels, and the Triangle Inequality},
  journal = {Journal of Mathematical Psychology},
  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.},
  volume = {52},
  number = {5},
  pages = {297-303},
  organization = {Max-Planck-Gesellschaft},
  school = {Biologische Kybernetik},
  month = sep,
  year = {2008},
  slug = {4785},
  author = {J{\"a}kel, F. and Sch{\"o}lkopf, B. and Wichmann, FA.},
  month_numeric = {9}
}