City, University of London Institutional Repository
Citation: Pothos, E. M., Busemeyer, J. R. and Trueblood, J. S. (2013). A quantum
geometric model of similarity. Psychological Review, 120(3), pp. 679-696. doi:
10.1037/a0033142
This is the unspecified version of the paper.
This version of the publication may differ from the final published
version.
Permanent repository link: https://openaccess.city.ac.uk/id/eprint/2250/
Link to published version: http://dx.doi.org/10.1037/a0033142
Copyright: City Research Online aims to make research outputs of City,
University of London available to a wider audience. Copyright and Moral
Rights remain with the author(s) and/or copyright holders. URLs from
City Research Online may be freely distributed and linked to.
Reuse: Copies of full items can be used for personal research or study,
educational, or not-for-profit purposes without prior permission or
charge. Provided that the authors, title and full bibliographic details are
credited, a hyperlink and/or URL is given for the original metadata page
and the content is not changed in any way.
City Research Online: http://openaccess.city.ac.uk/ publications@city.ac.uk
City Research Online
1 a quantum geometric model of similarity
A quantum geometric model of similarity
Emmanuel M. Pothos
1
, Jerome R. Busemeyer
2
, & Jennifer S.
Trueblood
3
Affiliations, correspondence: 1: Department of Psychology, City University, London, EC1V 0HB, UK,
emmanuel.pothos.1@city.ac.uk; 2: Department of Psychological and Brain Sciences, Indiana
University, Bloomington 47468, Indiana, USA, jbusemey@indiana.edu; 3: Department of Cognitive
Sciences, 3151 Social Sciences Plaza, University of California, Irvine, Irvine, CA 92697-5100, USA,
jstruebl@uci.edu
Word count: 10,672; Running head: a quantum geometric model of similarity
2 a quantum geometric model of similarity
Abstract
No other study has had as great an impact on the development of the similarity literature as that of
Tversky (1977), which provided compelling demonstrations against all the fundamental assumptions
of the popular, and extensively employed, geometric similarity models. Notably, similarity judgments
were shown to violate symmetry and the triangle inequality, and also be subject to context effects,
so that the same pair of items would be rated differently, depending on the presence of other items.
Quantum theory provides a generalized geometric approach to similarity and can address several of
Tversky’s (1997) main findings. Similarity is modeled as quantum probability, so that asymmetries
emerge as order effects, and the triangle equality violations and the diagnosticity effect can be
related to the context-dependent properties of quantum probability. We so demonstrate the
promise of the quantum approach for similarity and discuss the implications for representation
theory in general.
Keywords: similarity, metric axioms, symmetry, triangle inequality, diagnosticity, quantum
probability
3 a quantum geometric model of similarity
I. Introduction
The notion of similarity is, in equal measure, a famous hero and a notorious villain in psychology.
Across most areas of psychology, similarity plays a fundamental role (e.g., Goldstone, 1994; Pothos,
2005; Sloman & Rips, 1998), but equally its various formalizations have been the source of much
criticism and debate (e.g., Goodman, 1972). A popular approach to similarity is a geometric one,
according to which stimuli/ exemplars/ concepts are represented as points in a multidimensional
psychological space, with similarity being a function of distance in that space. This geometric
approach is exemplified in Shepard’s (1987) famous law of generalization, according to which
similarity is an exponentially decaying function of distance, and is heavily used in influential cognitive
models of categorization, such as exemplar and prototype theory. It is fair to say that cognitive
psychology cannot resist using a geometric approach to similarity.
This reliance on the geometric approach to similarity is surprising because it has been
subject to intense, and, in some cases, highly compelling criticisms. The most complete and
impactful expression of this criticism is that of Tversky (1977). Tversky’s work has had a profound
influence on the development of the similarity literature (over 2,200 citations), partly because his
objections to geometric similarity models concern the most basic properties of such models – the
metric axioms, that is, the fundamental properties that any similarity measure based on distance
must obey. Thus, if the metric axioms are shown to be inconsistent with psychological similarity,
then any distance model of similarity is essentially incorrect. Tversky’s (1977) demonstration is a rare
one, in that he has been able to convincingly argue against an entire modeling framework, rather
than particular models. This is because his arguments were not dependent on e.g. particular
parametric configurations, rather they concerned the fundamental properties of any model of
similarity based on distance in psychological space (though see Nosofsky, 1991, for a parametric
way to produce an asymmetric distance-based similarity measure). It is not surprising that Tversky’s
(1977) demonstrations have come to be accepted as the golden standard of key results any
successful similarity model should cover (Ashby & Perrin, 1988; Bowdle & Gentner, 1997; Goldstone
& Son, 2005; Krumhansl, 1978).
In brief, Tversky (1977) showed that similarity judgments violate minimality (identical
objects are not always judged to be maximally similar), symmetry (the similarity of A to B can be
different from that of B to A), and the triangle inequality (the distance between two points is always
shorter directly, than via a third point). Moreover, he showed that the similarity between the same
two objects can be affected by which other objects are present (called the diagnosticity effect). In
the typical tradition of his work, part of the reason why his findings have had the influence they did
is because they go against basic logic. For example, concerning his most famous result, violations of
symmetry, if similarity is determined by distance, then how could it be the case that the similarity/
distance between two objects depends on the order in which they are considered? Yet, when he
asked participants to choose between the statements ‘China is similar to Korea’ vs. ‘Korea is similar
to China’ (actually North Korea and Red China, but for simplicity we will just talk about Korea and
China’), 66 out of 69 participants selected the latter statement as more agreeable, implying that the
similarity of Korea to China (denoted as Sim(Korea, China)) is higher than that of China to Korea
(denoted as Sim (China, Korea)). Thus, this result provided a compelling (and retrospectively
intuitive) violation of symmetry in similarity. Tversky employed several other pairs of countries,
stimuli from other domains, and alternative procedures (see also Bowdle & Gentner, 1997,
4 a quantum geometric model of similarity
Catrambone, Beike, & Niedenthal, 1996, Holyoak & Gordon, 1983, Op de Beeck, Wagemans, &
Vogels, 2003, Ortony et al., 1985, and Rosch, 1975). Note that some researchers have questioned
the reality of asymmetries in similarity. For example, Gleitman et al. (1996) suggested that in
directional similarity statements, we cannot assume that, e.g., Korea gives rise to the same
representation in the target position, as it does in the referent position. But Gleitman et al.’s (1996)
analysis cannot explain why it is more intuitive to place, e.g., Korea in the referent, as opposed to
the target, position, the absence of asymmetries in some cases (Aguilar & Medin, 1999), and the
demonstration of similarity asymmetries with non-linguistic measures (Hodgetts & Hahn, 2012).
We will present what can be labeled a quantum similarity model. Quantum probability (QP)
theory is a theory for how to assign probabilities to events (for more refined characterizations see
e.g. Aerts & Gabora, 2005; Atmanspacher, Romer, & Wallach, 2006; Busemeyer & Bruza, 2012;
Khrennikov, 2010). QP theory is a geometric theory of probability. It is analogous to classical
probability theory, though QP theory and classical theory are founded from different sets of axioms
(the Kolmogorov and Dirac/ von Neumann axioms respectively) and so are subject to alternative
constraints. QP theory is based on linear algebra, augmented with a range of assumptions and
theorems (such as the Kochen-Specker theorem and Gleason’s theorem; Busemeyer & Bruza, 2012;
Hughes, 1989; Isham, 1989; Khrennikov, 2010). Note that a quantum approach to cognitive modeling
does not introduce assumptions regarding neural implementation and we are agnostic on this issue.
Specifically, operations which are quantum-like can emerge at the computational level from a
classical brain (Atmanspacher & beim Graben, 2007) and do not assume quantum neural
computations (this latter thesis is very controversial; Hameroff, 2007; Litt et al., 2006).
A unique feature of the quantum similarity model is that, whereas previous models would
equate objects with individual points or distributions of points, in the quantum model, objects are
entire subspaces of potentially very high dimensionality. This is an important generalization of
geometric models of similarity, as it leads to a naturally asymmetric similarity measure.
The quantum similarity model follows the recent interest in the application of quantum
probability (QP) theory to cognitive modeling. Applications of QP theory have been presented in
decision making (Blutner et al., in press; Busemeyer, Wang, & Townsend, 2006; Busemeyer et al.,
2011; Bordley, 1998; Lambert-Mogiliansky, Zamir, & Zwirn, 2009; Pothos & Busemeyer, 2009;
Trueblood & Busemeyer, 2011; Wang & Busemeyer, in press; Yukalov & Sornette, 2010), conceptual
combination (Aerts, 2009; Aerts & Gabora, 2005; Blutner, 2008; Bruza et al., under review), memory
(Bruza, 2010; Bruza et al., 2009), and perception (Atmanspacher, Filk, & Romer, 2004). Psychological
models based on quantum probability seem to work well (for overviews see Busemeyer & Bruza,
2009; Bruza et al., 2009; Khrennikov, 2004; Pothos & Busemeyer, in press) and add to the increasing
realization that the application of QP need not be restricted to physics. For example, QP has also
been applied to areas as diverse as economics (Baaquie, 2004) and information theory (Nielsen &
Chuang, 2010).
We first present QP theory and motivate our similarity model. Subsequently, we consider
three main results from Tversky (1977). Violations of symmetry, violations of the triangle inequality,
and the diagnosticity effect. Violations of symmetry provide and most compelling and intuitive
evidence against (simple) geometric representational models. Moreover, the diagnosticity effect is
obviously impossible to reconcile with similarity models based on distance alone, as it shows that
similarity judgments between the same two elements might be affected by the presence of other
elements. Note, we do not consider violations of minimality, i.e., the finding that naïve observers do
not always assign the maximum similarity rating for pairs of identical stimuli. Violations of minimality
