Contribution
Before this paper, clique topology (Giusti et al. 2015) had been used to detect geometric structure in neural correlations, and odour perception had been described as low-dimensional and qualitatively curved (Koulakov et al.’s “potato-chip”). This paper applies the Betti-curve test to the co-occurrence statistics of natural odours and to human perceptual ratings. It reports that both fit a 3D hyperbolic geometry and neither fits a uniform Euclidean cube. It also proposes that the matching geometry and dimension of stimulus and percept avoid the distortion a mismatch would force.
Key insight
If odours are produced together by branching biochemical pathways, their co-occurrence should be organised like the leaves of a tree. A tree embeds with low distortion in hyperbolic space, its leaves near the boundary. So the paper asks whether the rank order of odour correlations looks like a sample from near the boundary of a hyperbolic ball. Because Betti curves are invariant to any monotone transform of the correlations, the question is posed about geometry alone, without fixing how correlation maps to distance.
Assumptions
- Correlation is proximity. D_ij = −|C_ij|, the absolute Pearson correlation of two compounds’ concentrations across samples. Only the rank order is used, so any decreasing function of |C| gives the same result. The sign of a correlation is discarded.
- Uniform sampling in the candidate geometry. Euclidean: uniform in the d-cube. Hyperbolic: native ball of curvature ζ = 1, uniform angle, radial density ρ(r) ∝ sinh((d − 1)r) on [R_min, R_max], with distance by the hyperbolic law of cosines, cosh x = cosh r cosh r′ − sinh r sinh r′ cos Δθ.
- Multiplicative noise on model distances, D = D_geo (1 + ε N(0,1)), ε fitted per dataset (0.040–0.050 hyperbolic; 0.05–0.09 Euclidean). No noise for the perceptual set.
- Fitting. Parameters (R_max, R_min, ε, or d and ε) are tuned to the first integrated Betti value; the second and third are then the test.
- Perceptual distance is the Euclidean distance between 146-vectors of descriptor ratings (Dravnieks), not a correlation.
- Compounds whose concentrations were mostly zero were dropped.
Key results
- Natural odours (Fig. 2; tables S1–S4). All four datasets are
consistent with a 3D hyperbolic shell, R_max = 7 and R_min = 0.9 R_max
for all four: P > 0.25 (blueberry), > 0.21 (tomato), > 0.45 (mouse),
0.19 (strawberry), the minimum over Betti 2 and 3. The best Euclidean models (d = 8 or 10) are rejected: P < 0.03 (blueberry), < 0.003 (the rest). Shuffled concentrations fit random matrices (P = 0.4, 0.7, 0.9) and are rejected by the hyperbolic model (P < 0.01). The same holds with L1 distances between curves and with log concentrations. Hyperbolic dimension above 3 is not excluded, but 3 fits best (fig. S3).
- Embedding (Fig. 3). A nonmetric MDS modified for hyperbolic distance, with radii fixed and angles optimised, places the compounds approximately uniformly over the sphere. They do not cluster by functional group (fig. S5).
- Axes (Fig. 4). In the joint strawberry–tomato embedding (Procrustes aligned), directions for pleasantness, boiling point and acidity are fitted on two thirds of the compounds and tested on the rest. Pleasantness of single compounds, R = 0.66, P = 3 × 10⁻⁷; liking of tomato samples from a strawberry-trained axis, R = 0.34, P = 0.01; boiling point and acidity, P < 0.04. Pleasantness is predictable from the other two axes, since the shell is effectively 2D.
- Perception (Fig. 5; table S5). Euclidean spaces of every dimension fail the first integrated Betti value (P < 0.003). Hyperbolic balls of several dimensions match Betti 1, but only low dimensions also match Betti 2. 3D is best, and d ≥ 9 is rejected (P < 0.034). Betti 3 is about zero. The biphasic curves are reproduced by resampling the hyperbolic MDS embedding, whose points fill half the ball (L1 P = 0.32 and 0.20 hyperbolic, P = 0 and 0.06 Euclidean). Hyperbolic MDS distances correlate better with perceptual distances than Euclidean ones (fig. S6).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Rank-ordered concentration correlations in four natural odour sources match a 3D hyperbolic shell and not a uniform Euclidean cube | moderate (four datasets, shuffle control, two scoring rules; the only flat null is a uniform cube) | Fig. 2; tables S1–S4 |
| C2 | All four sources share one hyperbolic radius | weak (a single grid-fitted shell, R_max = 7, R_min = 0.9 R_max, reported identical; how finely R was searched is not stated) | Methods |
| C3 | Human odour-descriptor profiles match a full 3D hyperbolic ball and no Euclidean space | moderate (one dataset of 127 odorants) | Fig. 5; table S5 |
| C4 | Pleasantness, boiling point and acidity are linear axes of the fitted fruit-odour space, with held-out validation | moderate for pleasantness of single compounds; weak for mixtures (R = 0.34) | Fig. 4 |
| C5 | Natural odour space is hierarchical, a tree whose leaves are the compounds | argued, not tested; no tree is inferred and compounds do not cluster by chemistry | Fig. 1; Discussion |
| C6 | Matching stimulus and perceptual geometry minimises perceptual distortion | conjecture | Discussion |
| C7 | Perceptual spaces in general are hyperbolic, because of hierarchical networks and saturating neurons | conjecture, supported only by citations to visual, haptic and auditory work | Discussion |
Method
Betti curves of the order complex of a similarity matrix (Giusti et al.) are compared with those of matrices built from points sampled in each candidate geometry: 300 samples per candidate, P values as two-tailed percentiles of the data’s integrated Betti value, or of its L1 distance to the model’s mean curve. Model parameters are tuned on Betti 1 and tested on Betti 2 and 3. Visualisation is by MATLAB’s nonmetric MDS with the distance replaced by the hyperbolic one, initialised from the fitted shell, radii held fixed.
Concepts
- olfactory space: here, the space whose points are monomolecular compounds (or odour mixtures) and whose distances are set by co-occurrence across natural samples. The perceptual space is the same for descriptor profiles.
- integrated Betti value: the area under a Betti curve, the number of m-dimensional cycles plotted against edge density.
- hyperbolic shell: the ball model sampled only between R_min and R_max, standing for the leaves of a tree.
- Venn-to-half-space map: a disc of centre (x, y) and radius ρ becomes the point (x, y, ρ) of the upper half-space, so larger, more inclusive sets sit higher. Cited to Krioukov et al. (ref. 7), not proved here.
Connections
- Krioukov et al. (2010), filed in the same batch (LIT-865, NOTE-680, THEORY-188), is ref. 7: the ball model, the radial density sinh((d − 1)r), the law of cosines, and the reading of a boundary shell as the leaves of a tree all come from there. Ref. 21 is Boguñá, Papadopoulos & Krioukov on Internet routing.
- Zhang, Rich, Lee & Sharpee (2022) (LIT-876, NOTE-668, THEORY-192) is the same lab applying the same test to CA1 place cells, with a full ball rather than a shell. Its reading names this paper as prior art. The confound named there, that the metric is not separated from the distribution of points, holds here too, and here in a sharper form.
- Sala et al. (2018) (LIT-874, THEORY-196): its subject is the precision cost of embedding a known tree. This paper embeds no tree, so those tradeoffs are not engaged.
- In the record’s holdings nothing else concerns olfaction, odour statistics or perceptual geometry.
Bearing on the record
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QUESTION-025. Indirect, but closer than the hippocampal paper, for two reasons.
- The data here are co-occurrence statistics: the correlation of components across natural samples, the analogue of word co-occurrence. They are reported to be non-Euclidean in a rank-invariant sense.
- Linear attribute directions are nevertheless read out of the fitted curved space.
Neither result answers the question. The attributes (pleasantness, boiling point, acidity) are continuous and not hierarchical. No implication structure among binary attributes is posited or measured. The axes are fitted in the angular coordinates of a shell that is close to a sphere, not in the hyperbolic metric. The paper’s Fig. 1 argument does bear on the lattice side of the question: a family of sets (a formal context’s extents) maps to points of a hyperbolic half-space with inclusion as height. That suggests implication among attributes would appear as radial depth rather than as a sum of attribute vectors, against THEORY-185’s premise. The argument is cited from Krioukov et al., not shown here, and it is not tested.
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It is the source of THEORY-191.
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No CLAIM in the record rests on odour or perceptual geometry, so it grounds none. It gives no instruction for ML practice, so nothing goes to the anthology.
Limitations
- The only flat null is a uniform cube. On a shell of fixed radius, hyperbolic distance is a strictly increasing function of the angle between points, so its rank order is that of points on a 2-sphere. The fitted shell is thin (6.3 ≤ r ≤ 7, with most mass near 7 under the sinh 2r density). In my simulation of it, hyperbolic and spherical angular distances have a Spearman correlation of about 0.91. The radial spread and the noise carry what difference remains. Points on a sphere, or a sphere with a per-point offset, in Euclidean 3-space were not tested, and the “essentially 2D” space of the Results says as much. The natural-odour result therefore does not cleanly separate a hyperbolic metric from a spherical arrangement in flat space. This is my observation; the paper does not discuss it. The perceptual result (full ball, R_max = 1.6) is not subject to it in this form.
- No hierarchy is recovered. The tree is the motivation. Compounds are spread uniformly over the shell and do not cluster by chemistry, so the data show no branching structure.
- Small samples. 45–78 compounds over 50–101 samples per source; one perceptual dataset of 127 odorants, which the authors say samples perceptual space non-uniformly.
- Fitting freedom. The hyperbolic model has R_max, R_min and ε; the Euclidean, d and ε. Both are tuned on Betti 1 before testing on Betti 2 and 3. The grid over R and the per-dataset P values are in the supplement, which was not read.
- Sign discarded. |C| treats anticorrelated compounds as close, which fits co-production only if anticorrelation also means a shared pathway.
- Separate spaces. Fruit and mouse sets share no compounds, so their common radius is a fitted coincidence of parameters, not one embedding.
- The perceptual-distortion and cross-modal claims (C6, C7) are discussion only.
Open questions
- Does a flat model with points on or near a 2-sphere, or a flat space with an exponentially distributed scale coordinate, fit the odour Betti curves as well? Rejecting both would make the hyperbolic metric necessary rather than sufficient.
- Can a tree be inferred from odour co-occurrence, by hierarchical clustering or an ultrametric fit, and does it match known biosynthetic pathways? That would test C5 directly.
- Does a larger, pooled odour set fill the interior of the ball, as the Discussion suggests, and is the dimension still 3?