# The differential geometry of perceptual similarity

arXiv: https://arxiv.org/abs/1708.00138 · HTML: https://mlsanity.com/paper/1708.00138/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Aug 2017.

## Built on by (most cited first, 2 of 2)

- [Neural Networks as Paths through the Space of Representations](https://mlsanity.com/paper/2206.10999/) (2022)
- [Geometry matters: insights from Ollivier Ricci Curvature and Ricci Flow into representational alignment through Ollivier-Ricci Curvature and Ricci Flow](https://mlsanity.com/paper/2501.00919/) (2025)

## Abstract (arXiv)

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.