Perceptual colour space is not even Riemannian, because diminishing returns cannot exist in a Riemannian geometry (Bujack 2022)

Bujack and colleagues (PNAS 2022) report a result that undercuts a century-old assumption in colour science: perceptual colour space cannot be described by a Riemannian metric at all. Their finding is that "a Riemannian metric overestimates the perception of large color differences" — because perception exhibits diminishing returns (a large jump is perceived as less than the sum of its small steps), and diminishing returns cannot exist in a Riemannian geometry, where distances are integrals of a local metric along paths. The Helmholtz–Schrödinger tradition of modelling colour discrimination with a local (Riemannian) line element is therefore insufficient as a global model of perceived difference.

This matters to the vault beyond colorimetry. It is the load-bearing caveat on the CIELAB↔natural-gradient bridge (observation-cielab-nonuniformity-bridges-gradient-geometry-not-sampling-artifact): Amari's natural gradient (claim-amari-1998-natural-gradient-fisher-steepest-descent) rests on a Riemannian (Fisher information) metric over the parameter manifold, whereas the perceptual space CIELAB approximates is not even Riemannian. So CIELAB and natural gradient share the Euclidean→non-Euclidean corrective move but not the same underlying geometry — evidence for the "analogy, not one object" side of question-gradient-geometry-one-object-or-three-analogies. It also raises a live research thread: whether a "diminishing returns" failure of the Riemannian frame has an analogue for parameter manifolds in gradient geometry.