A Quantum Geometric Model of Color Similarity Judgements

Since Tversky (1977) argued that similarity judgments violate the three metric axioms, asymmetrical similarity judgments have been offered as particularly difficult challenges for standard, geometric models of similarity, such as multidimensional scaling. According to Tversky (1977), asymmetrical similarity judgments are driven by differences in salience or extent of knowledge. However, the notion of salience has been difficult to operationalize to different kinds of stimuli, especially perceptual stimuli for which there are no apparent differences in extent of knowledge. To investigate similarity judgments between perceptual stimuli, across three experiments we collected data where individuals would rate the similarity of a pair of temporally separated color patches. We identified several violations of symmetry in the empirical results, which the conventional multidimensional scaling model cannot readily capture. Pothos et al. (2013) proposed a quantum geometric model of similarity to account for Tversky’s (1977) findings. In the present work, we developed this model to a form that can be fit to similarity judgments. We fit several variants of quantum and multidimensional scaling models to the behavioral data and concluded in favor of the quantum approach. Without further modifications of the model, the quantum model additionally predicted violations of the triangle inequality that we observed in the same data. Overall, by offering a different form of geometric representation, the quantum geometric model of similarity provides a viable alternative to multidimensional scaling for modeling similarity judgments, while still allowing a convenient, spatial illustration of similarity.