Imagine you’re trying to compare the similarity between different colors – sounds easy, right? Well, it turns out that our judgments of color similarity can be quite tricky to model. Standard geometric models of similarity struggle with asymmetrical judgments, where one color may appear more similar to another than vice versa. To tackle this challenge, researchers have proposed a quantum geometric model of similarity, which takes into account factors like salience and extent of knowledge. In this study, the researchers built upon this model to create a more comprehensive framework that can accurately capture similarity judgments. They collected data on individuals rating the similarity of pairs of temporally separated color patches and found several violations of symmetry in the results. The conventional multidimensional scaling model couldn’t adequately explain these findings, but the quantum approach offered a better fit. Interestingly, the quantum model also predicted violations of the triangle inequality, further supporting its effectiveness. By providing a new way of representing similarity, this quantum geometric framework offers an alternative to traditional models while still allowing for a convenient spatial visualization. Curious to dive deeper into this fascinating research? Check out the full article!

Abstract
Since Tversky argued that similarity judgments violate the three metric axioms, asymmetrical similarity judgments have been particularly challenging for standard, geometric models of similarity, such as multidimensional scaling. According to Tversky, asymmetrical similarity judgments are driven by differences in salience or extent of knowledge. However, the notion of salience has been difficult to operationalize, especially for 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. proposed a quantum geometric model of similarity to account for Tversky’s findings. In the present work, we extended this model to a more general framework that can be fit to similarity judgments. We fitted 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 best-fit 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 framework of similarity provides a viable alternative to multidimensional scaling for modeling similarity judgments, while still allowing a convenient, spatial illustration of similarity.

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