Source
Zhou, Smith & Sharpee (2018), Science Advances 4(8):eaaq1458, LIT-872, Figs. 2 and 5 and Materials and Methods, as read in NOTE-672.
What was actually shown
Each pair of compounds gets the distance −|corr| of their concentrations across natural samples. Clique topology turns the matrix into Betti curves, the counts of 1-, 2- and 3-cycles as the edge threshold sweeps. The curves depend only on the rank order of the correlations. For each candidate geometry, as many points as compounds are sampled, their distances get 4–5% multiplicative noise, and 300 such samples give a distribution against which the data’s integrated Betti values are scored. Parameters are tuned on the first curve and tested on the second and third.
For four sources (45–78 compounds over 50–101 samples each), a 3D hyperbolic ball sampled in the shell 6.3 ≤ r ≤ 7 fits all three curves (P > 0.19 throughout), with one radius for all four. The best uniform Euclidean cubes (d = 8 or 10) are rejected (P < 0.03, and < 0.003 for three of the four). Shuffled data look like random matrices and are rejected by the hyperbolic model. For Dravnieks’ 146-descriptor profiles of 127 odorants, a full 3D hyperbolic ball fits the first two curves, and no Euclidean dimension matches the first (P < 0.003). Hyperbolic dimension 9 and above is rejected there. The test could reject: it rejected the cubes, the shuffles and high-dimensional hyperbolic balls.
What this does not say
- Not that the odour metric is hyperbolic as against a flat sphere. On a shell of fixed radius, hyperbolic distance increases strictly with the angle between points, so the rank order of distances is that of points on a 2-sphere. The fitted shell is thin. In a simulation of it made for this record, hyperbolic and angular distances have a Spearman correlation of about 0.91. A sphere, or a near-sphere, in Euclidean space was not among the nulls; only uniform cubes were. The perceptual result, on a full ball, is not open to this objection in this form, but it shares the general one: a uniform cube is the only flat null, so the hyperbolic metric is not separated from a non-uniform distribution of points.
- Not that odour space is a tree. The hierarchy of biochemical pathways motivates the model and is not recovered: compounds spread uniformly over the shell and do not cluster by chemical class.
- Not that the perceptual and stimulus spaces are the same space. They share a dimension and a curvature sign, but with different radii and a shell against a full ball, and no mapping between them is fitted.
- Not general. Four odour sources, one perceptual dataset; the title names them. Whether the match replicates, and whether it survives flat near-spherical nulls, is QUESTION-153. Nothing here says other sensory or conceptual spaces are hyperbolic, or that a co-occurrence embedding of words would be.