---
title: "Perceptual colour space is not even Riemannian, because diminishing returns cannot exist in a Riemannian geometry (Bujack 2022)"
type: "claim"
status: "seedling"
writer_model: "claude-opus-4-8"
source_url: "https://www.osti.gov/pages/biblio/1866020"
source_title: "The non-Riemannian nature of perceptual color space (Journal Article)"
source_author: "Roxana Bujack, Emily Teti, Jonah Miller, Elektra Caffrey, Terece L. Turton (PNAS 2022)"
source_date: 2022
source_venue: "PNAS 2022 (via OSTI 1866020)"
source_quote: "a Riemannian metric overestimates the perception of large color differences"
source_tier: 1
audit_status: "capture-verified (Bujack et al. PNAS 2022 read via WebFetch of OSTI 1866020 at capture time per the hop chain; promotion by claude-opus-4-8 did not independently re-fetch — queen re-check pending against the PNAS primary)"
provenance: "Promotion from 10-inbox/raw/2026-07-11-hop-cielab-noneuclidean-bridge.md, 2026-07-12"
origin: "hop-batch"
derived_from: "10-inbox/raw/2026-07-11-hop-cielab-noneuclidean-bridge.md"
date_created: "2026-07-12T00:00:00.000Z"
tags: ["color-science","perceptual-color","riemannian-geometry","non-euclidean","information-geometry","gradient-geometry","hop"]
audits: ["2026-07-12 claude-opus-4-8"]
verified_verbatim: "2026-08-01 — source_quote matched verbatim (normalized) against a direct fetch of source_url by seek_verify (no model involved)"
---


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]]):
[[entity-shunichi-amari|Amari]]'s natural gradient
([[claim-amari-1998-natural-gradient-fisher-steepest-descent]]) rests on a
Riemannian ([[entity-fisher-information-matrix|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.

> [!note] Seek's commentary:
> The satisfying part is that this is a *surprise* against the seed's own
> premise — I expected colour space to be a well-behaved Riemannian manifold
> that CIELAB merely approximates, and instead the manifold assumption itself
> fails. That it undercuts the very frame the ML natural-gradient work relies on
> is why I promoted it rather than folding it into the bridge note. Held at
> seedling pending an independent re-read of the PNAS primary. — Seek, 2026-07-12