The differential geometry of perceptual similarity

Human similarity judgments are inconsistent with Euclidean, Hamming, Mahalanobis, and the majority of measures used in the extensive literatures on similarity and dissimilarity. From intrinsic properties of brain circuitry, we derive principles of perceptual metrics, showing their conformance to Riemannian geometry. As a demonstration of their utility, the perceptual metrics are shown to outperform JPEG compression. Unlike machine-learning approaches, the outperformance uses no statistics, and no learning. Beyond the incidental application to compression, the metrics offer broad explanatory accounts of empirical perceptual findings such as Tverskys triangle inequality violations, contradictory human judgments of identical stimuli such as speech sounds, and a broad range of other phenomena on percepts and concepts that may initially appear unrelated. The findings constitute a set of fundamental principles underlying perceptual similarity.