PaperlayerView exact revision on arXiv

Source-linked AI summary

Geometric organization of olfactory descriptor data in the Poincaré disk

Aniss Aiman Medbouhi, Farzaneh Taleb, Giovanni Luca Marchetti, Danica Kragic

arXiv:2609.09573v1cs.CEcs.LGstat.AP

TL;DR

Low-dimensional organization of olfactory descriptor profiles remains unclear. This study applies hyperbolic multidimensional scaling to two descriptor datasets and finds complementary radial and angular organization in their embeddings.

  • Problem

    Whether olfactory descriptor profiles have an interpretable low-dimensional representation remains unclear, and no widely accepted mapping of olfactory perceptual space exists.

    Source: §1 · p. 2

  • Method

    The study applies hyperbolic metric multidimensional scaling to continuous human odor ratings and binary molecular descriptor data, then analyzes radial and angular organization.

    Source: §1 · p. 3, §2.4.1 · p. 4

  • Results

    The embeddings showed complementary organization: Sagar profile entropy decreased with radius, while GoodScents entropy relationships depended on definition and related binary descriptors occupied coherent disk regions.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Two-dimensional hyperbolic mapping provides a descriptive framework separating global descriptor-profile organization from descriptor-specific gradients and categorical odor-quality structure.

    Source: §1 · p. 3

    The analysis uses descriptor data and does not directly test neural mechanisms of olfactory coding.

    Source: §4.2 · p. 15

Abstract

from arXiv · show

Odor quality is commonly represented using high dimensional descriptor profiles, yet their low dimensional organization remains unclear. We investigated whether a two-dimensional hyperbolic embedding can provide an interpretable representation of this structure. We applied hyperbolic metric multidimensional scaling to two complementary datasets: 480 Sagar rating profiles from three participants rating 160 odorants on 15 continuous descriptors, and 4983 GoodScents--Leffingwell molecules annotated with 138 binary descriptors. The embeddings substantially preserved pairwise descriptor distances, supporting subsequent analyses of radial and angular organization. In Sagar, rating profile entropy was strongly and negatively associated with hyperbolic radius, with diffuse profiles closer to the center and concentrated profiles closer to the boundary. This radial organization emerged primarily at the level of the full descriptor profile, rather than any individual descriptor, and remained robust across alternative descriptor representations, participant specific analyses, and averaged ratings. Sweet, musky, fruity, pleasantness showed the strongest directional trends. In GoodScents--Leffingwell, active label entropy, reflecting descriptor multiplicity, increased with radius, whereas orthogonalized descriptor entropy, reflecting spread across orthogonal modes, decreased with radius. Related binary descriptors occupied coherent localized high-density regions. These findings reveal complementary radial and angular organization in the hyperbolic representation of olfactory descriptor data. They support hyperbolic mapping as an interpretable descriptive framework in which radius summarizes global profile properties, while the angular component captures continuous descriptor gradients and categorical organization.

1 Introduction

The study uses two-dimensional hyperbolic mapping to examine how descriptor profiles are organized radially and angularly. It reports a strong radial association for Sagar rating-profile entropy and complementary entropy-related and descriptor-region organization in GoodScents–Leffingwell.

Motivation and contribution

  • The study addresses the lack of an agreed olfactory descriptor-space organization by extending prior preliminary Sagar analyses into a broader, statistically validated framework.It replaces contrastive learning with hyperbolic metric MDS and adds a radial–angular decomposition, distance-preservation evaluation, complementary visualizations, restricted permutation tests, and robustness analyses.Source: §1 · pp. 2–3

Approach

  • Hyperbolic metric MDS embeds observations in the two-dimensional Poincaré disk by minimizing differences between descriptor-space and hyperbolic pairwise distances.The representation is intended to aid visualization and interpretation of center-to-boundary variation and angular organization, not to estimate intrinsic dimensionality or neural geometry.Source: §1 · pp. 2–3

Data and questions

  • The study analyzes continuous ratings from three Sagar participants and binary expert annotations for several thousand GoodScents–Leffingwell molecules.It evaluates pairwise distance preservation and asks whether radius tracks global profile properties, descriptors show directional or localized organization, and patterns persist across initializations and alternative representations.Source: §1 · p. 3

Results

  • Sagar rating-profile entropy was strongly negatively associated with radius, with diffuse profiles nearer the center and concentrated profiles nearer the boundary.The association was stronger than that of any individual descriptor and persisted across alternative descriptor representations, participant-specific analyses, and averaged ratings; sweet, musky, fruity, and pleasantness showed consistent directional trends.Source: §1 · p. 3
  • In GoodScents–Leffingwell, radius related to entropy properties in ways that depended on the entropy definition, while related binary descriptors occupied coherent disk regions.Together, the findings support hyperbolic mapping as a descriptive framework separating global profile organization from descriptor-specific gradients and categorical odor structure.Source: §1 · p. 3

2 Materials and methods

This section describes methods for embedding two complementary odor-descriptor datasets in a two-dimensional Poincaré disk and evaluating descriptor-distance preservation and radial and angular organization. It covers entropy-based and continuous-descriptor analyses, permutation tests for radial associations, and density mapping to assess binary descriptors’ spatial concentration.

2.2 Data sources

  • Sagar contributes 480 subject–odorant observations with 15 shared continuous ratings, while GoodScents–Leffingwell contributes 4,983 molecules with 138 binary odor descriptors.Sagar observations span 195 unique CIDs; GoodScents–Leffingwell is a complementary, non-repeated-measures dataset.Source: §2.2 · pp. 3–4

2.3 Hyperbolic metric multidimensional scaling

  • Hyperbolic metric multidimensional scaling learns two-dimensional Poincaré disk coordinates by minimizing discrepancies between input and embedded pairwise distances.Coordinates are initialized using a hyperbolic analogue of a Gaussian distribution and optimized with Riemannian Adam (Becigneul & Ganea, 2019).Source: §2.3 · p. 4
  • The study treats radius as center-to-boundary position and angle as direction, interpreting only relative directional or regional structure because absolute orientation can rotate or reflect across initializations.Hyperbolic distances expand near the disk boundary, allowing mutually separated observations to occupy large radii.Source: §2.3 · p. 4

2.4 Evaluation of embedding quality and geometric organization

  • The analysis first checks whether hyperbolic distances preserve input descriptor geometry, then tests how entropy, continuous ratings, and binary labels organize radially and angularly.Directional trends are analyzed for continuous ratings, while high-density regions are examined for binary descriptors; distance correlations are treated as quality-control metrics, not hypothesis tests.Source: §2.4 · p. 4

2.4.1 Entropy

  • Entropy is dataset-specific: Sagar’s rating profile entropy summarizes spread across graded descriptors, whereas GoodScents–Leffingwell active label entropy summarizes the number of active labels.Orthogonalized descriptor entropy tests robustness to replacing descriptors with orthogonal modes of variation; it measures spread across modes, not descriptor multiplicity.Source: §2.4.1 · pp. 4–5

2.4.2 Geometric organization.

  • Radial analyses correlate embedding radius with entropy or continuous ratings, while tangent-space regressions summarize global directional trends in Sagar descriptors.Pearson and Spearman radial correlations are averaged over random seeds; directional trends use the fitted plane’s steepest-increase direction and R^2, and may miss nonlinear, radial, or multi-cluster patterns. Restricted Sagar permutations preserve dependence through within-subject shuffling or odorant-profile blocks.Source: §2.4.2 · pp. 6–7
  • For radial tests, Sagar permutations swap whole odorant profiles only among matching subject-observation patterns, while GSLF shuffles scalar values across molecules with the embedding fixed.The radial statistic is the mean radius–value correlation over seeds, and its permutation test is two-sided.Source: §2.4.2 · p. 7
  • For continuous Sagar descriptors, restricted permutations refit tangent-space regressions to test whether global directions explain descriptor values better than permuted values.Angular permutation tests are one-sided because larger R2 indicates stronger directional organization.Source: §2.4.2 · pp. 7–8
  • For binary descriptors, hyperbolic kernel-density estimates identify disk regions containing a target fraction of descriptor mass, revealing coherent and potentially distinct angular sectors.The target mass is computed using the Poincaré disk’s hyperbolic area element; binary labels are analyzed by spatial concentration rather than as continuous perceptual gradients.Source: §2.4.2 · p. 8

3 Results

The Sagar results associate radius with whole-profile entropy and angular position with continuous descriptor gradients, with these patterns persisting across robustness analyses. In GoodScents–Leffingwell, radius tracks two distinct entropy properties, while angular regions organize binary odor families and subcategories.

3.1.1 Main results

  • −0.77±0.05 (Pearson) and −0.72±0.05 (Spearman) were the Sagar radius–entropy correlations; high-entropy profiles lay nearer the center and low-entropy profiles nearer the boundary.Intensity was the only individual descriptor with a non-negligible radial association (Pearson 0.42 ± 0.05; Spearman 0.38 ± 0.05), weaker than the whole-profile entropy association.Source: Figure 1 · p. 9, Table 1 · p. 9, §3.1.1 · p. 9
  • Mean directional R2 values of 0.40–0.51 for sweet, musky, fruity, and pleasantness showed strong angular organization in Sagar.All reported angular trends passed both restricted permutation tests (pWS < 0.001 and pOB < 0.001).Source: §3.1.1 · pp. 9–10, Figure 2 · p. 10, Table 2 · p. 10

3.1.2 Ablation and robustness analyses of the radial entropy organization

  • Across Sagar ablations, radius–entropy associations remained negative and substantial, including when intensity was excluded, descriptors were pruned, or entropy used orthogonalized descriptors.With intensity excluded from both training and entropy computation, correlations were −0.74±0.05 (Pearson) and −0.69 ± 0.06 (Spearman); orthogonalized entropy correlations were −0.69±0.07 (Pearson) and −0.79±0.09 (Spearman). Across configurations, distance correlations remained around 0.81–0.83, and all radius–entropy tests had pWS < 0.001 and pOB < 0.001.Source: §3.1.2 · pp. 10–11, Table 3 · p. 11

3.1.3 Subject-level and averaged-rating robustness analyses

  • Within-subject Pearson radius–entropy correlations were −0.72 ± 0.07, −0.86 ± 0.03, and −0.61 ± 0.06 for subjects 1, 2, and 3; averaged ratings yielded −0.82±0.08.Subject 2 had the strongest association. Pleasantness, sweet, musky, fruity, floral, burnt, and bakery had mean directional R2 > 0.2 and permutation p-values below 0.001 in all three subjects and the averaged-rating embedding, though directional strength varied among individuals.Source: Table 4 · p. 11, §3.1.3 · pp. 11–12, Figure 3 · p. 12

3.2 GoodScents–Leffingwell (GSLF)

  • In GSLF, active label entropy correlated positively with radius (Pearson 0.87±0.03; Spearman 0.88±0.03), while orthogonalized descriptor entropy correlated negatively (Pearson −0.88±0.01; Spearman −0.92 ± 0.02).The active-label association remained high after descriptor pruning (Pearson and Spearman both 0.78 ± 0.03); all three associations had pmol < 0.001. Active label entropy reflects descriptor multiplicity, whereas orthogonalized entropy reflects spread across orthogonal modes.Source: Table 5 · p. 13, §3.2 · p. 13
  • GSLF angular regions separated floral, meaty, and ethereal descriptor families and organized fruit-related subcategories, with bergamot, lemon, and coconut farther from the main fruity region.Related descriptors tended to occupy nearby regions, while individual fruit descriptors remained locally distinguishable.Source: §3.2 · pp. 13, 15, Figure 6 · p. 14, Figure 7 · p. 14

4 Discussion

The discussion frames the Poincaré-disk embedding as a representation with complementary radial organization of global descriptor-profile properties and angular organization of descriptor trends and categories. It also emphasizes limitations of the current data and identifies directions for testing the interpretation and scope of these patterns.

4.1 Summary and interpretation

  • In Sagar, radius tracked rating-profile entropy across robustness analyses, with diffuse profiles nearer the center and concentrated profiles nearer the boundary, supporting a global rather than descriptor-specific interpretation.The authors present hyperbolic mapping as a descriptive framework in which radius summarizes global profile properties.Source: §4.1 · p. 15
  • Sweet, musky, fruity, pleasantness, and decayed were summarized by directional trends; these patterns also appeared in subject-specific and averaged analyses, with strength varying across subjects.The discussion notes that pleasantness retains an established perceptual dimension while radial organization reflects the descriptor profile as a whole.Source: §4.1 · p. 15
  • In GSLF, active label entropy increased with radius while orthogonalized descriptor entropy decreased, and related binary descriptors occupied localized high-density regions in distinct angular sectors.The two entropy measures capture different properties: descriptor multiplicity and spread across orthogonal modes, respectively.Source: §4.1 · p. 15

4.2 Limitations and future work

  • The analysis did not test neural mechanisms, and its continuous-rating dataset had few subjects; larger, more diverse samples and richer expert vocabularies could test pattern consistency and hierarchical odor-category relations.The discussion proposes explicit multilevel taxonomies as a setting in which hyperbolic embeddings may be more informative.Source: §4.2 · p. 15
  • Binary-descriptor KDE regions are qualitative evidence of coherent label locations, not formal category-separation tests; future work could quantify concentration, overlap, and separation, and integrate molecular structure.A joint hyperbolic framework could examine how chemical similarity relates to radial entropy organization and angular odor-category structure.Source: §4.2 · p. 15
  • Without behavioral, cognitive, personality, or clinical measures, subject differences indicate rating patterns rather than specific mechanisms, and entropy’s psychological meaning remains tentative because participants did not judge it directly.The discussion proposes direct judgments of clarity, complexity, ambiguity, and confidence, alongside psychophysical and clinical measures, to test possible interpretations.Source: §4.2 · pp. 15–16

Supplementary material · A Hyperbolic geometry and optimization details

The supplementary material documents the geometry and optimization underlying the hyperbolic metric MDS model for reproducibility; the main text contains the concepts needed to understand the radial and directional analyses.

Supplementary material

A Hyperbolic geometry and optimization details

A.1 Geometry of the Poincaré ball

The section defines the Poincaré ball model and its position-dependent Riemannian metric, then describes the geodesic distance and radial geometry used to represent hyperbolic space.

A.1 Geometry of the Poincaré ball

  • The Poincaré ball models n-dimensional, curvature −1 hyperbolic space as a Euclidean ball with a position-dependent scaling of the Euclidean inner product.The metric tensor and corresponding tangent-vector norm are defined using this scaling.Source: §A.1 · p. 19
  • Geodesic distance is the length of the shortest path between two points in the ball and has an explicit expression.It is defined by minimizing path length over smooth curves joining the points.Source: §A.1 · p. 19
  • Equal Euclidean displacements correspond to increasingly large hyperbolic distances near the disk boundary.The section also defines the hyperbolic radius of a point.Source: §A.1 · p. 19

A.2 Möbius addition

Möbius addition provides a common operation for expressing the exponential map, logarithmic map, and parallel transport; its additive inverse is the Euclidean negative.

A.2 Möbius addition

  • The exponential map, logarithmic map, and parallel transport can all be written using Möbius addition.The passage introduces the operation for x,y ∈ P^n but does not include its defining formula.Source: §A.2 · p. 19
  • The additive inverse of x under Möbius addition is its Euclidean negative, −x.Source: §A.2 · p. 19

A.3 Tangent spaces and geometric maps

Tangent spaces in the Poincaré manifold are vector spaces identified with R^n, while their Riemannian inner products depend on location. Exponential and logarithmic maps connect these tangent spaces to geodesics and hyperbolic distance, and origin-based expressions support the continuous-rating analysis.

A.3 Tangent spaces and geometric maps

  • The tangent space T_zP^n is identifiable with R^n, but its inner product depends on z through the Riemannian metric.Source: §A.3 · p. 20
  • The exponential map sends a tangent vector from z to the manifold along the geodesic starting at z in that vector’s direction.Source: §A.3 · p. 20
  • The logarithmic map returns the tangent vector at z pointing along the geodesic from z to a point y, and its Riemannian norm equals their hyperbolic distance.Source: §A.3 · p. 20
  • The tangent vector’s Euclidean norm generally differs from hyperbolic distance because the tangent-space metric is scaled by λ_z.Source: §A.3 · p. 20
  • Origin-based expressions are used for tangent-space analysis of continuous descriptor ratings in the main text.Source: §A.3 · p. 20

A.4 Parallel transport

Parallel transport moves tangent vectors between distinct tangent spaces along geodesics while preserving the Riemannian geometry, enabling their use during manifold optimization.

A.4 Parallel transport

  • Parallel transport moves a tangent vector along a geodesic between distinct tangent spaces while preserving the Riemannian geometry.Tangent vectors at different points cannot be directly added because their tangent spaces differ.Source: §A.4 · p. 20
  • For z,y ∈Pn and v ∈TzPn, transport along the geodesic from z to y is given by a formula (Ganea et al., 2018; Becigneul & Ganea, 2019).Source: §A.4 · p. 20
  • The gyration operator in the hyperbolic gyrovector formalism is defined by Ungar (2009).Source: §A.4 · p. 20
  • During optimization, parallel transport transfers the first moment estimate between successive embedding positions.Source: §A.4 · p. 20

A.5 Embeddings initialization

Embeddings were initialized by mapping Gaussian samples from the tangent space at the Poincaré disk’s origin into the disk via the exponential map.

A.5 Embeddings initialization

  • Initialization used the pseudo-hyperbolic Gaussian, formed by pushing standard Gaussian samples from the origin’s tangent space through the exponential map.Source: §A.5 · p. 20
  • This procedure produced valid initial points inside the open unit disk.Source: §A.5 · p. 20
  • Independent samples were used for each random initialization, with σinit = 0.1 in all experiments.Source: §A.5 · p. 20

A.6 Riemannian Adam optimization

The embedding is optimized on the Poincaré disk with a Riemannian Adam method that combines manifold-specific operations with bias correction and projection for numerical stability. Training uses fixed hyperparameters and dataset-specific batching, and the loss reaches a stable plateau by 1,000 epochs.

A.6 Riemannian Adam optimization

  • Riemannian Adam uses closed-form Poincaré gradients, exponential maps, and parallel transport, with standard Adam bias correction and projection into the open unit disk for stability.The method follows Becigneul & Ganea (2019), Kingma & Ba (2015), and Kochurov et al. (2020).Source: §A.6 · p. 21
  • Each embedding point uses a single scalar second-moment estimate rather than coordinate-wise estimates, while parallel transport preserves the first moment across tangent spaces.The previous first moment is transported to the current tangent space between iterations.Source: §A.6 · p. 21
  • Training used η = 0.1, β1 = 0.9, β2 = 0.999, ε = 10−8, and εproj = 10−5; Sagar used full batches, while GSLF used mini-batches of 195 molecules.All configurations ran for 1,000 epochs, and each optimization loss reached a stable plateau by the end.Source: §A.6 · p. 21

B Additional results

The additional results provide full radial and angular descriptor analyses for the Sagar union embedding, alongside a representative visualization of directional organization.

B Additional results

  • The full radial descriptor results report Pearson and Spearman correlations with hyperbolic radius, averaged over 10 random seeds, with pWS and pOB permutation p-values.pWS denotes within-subject permutation; pOB denotes odorant-block permutation.Source: Table S1 · p. 22
  • The full angular results report directional R2 for the 15 descriptors; a representative embedding shows rating-colored panels and fitted tangent-space directions for initialization m=5.Directional R2 quantifies how well each descriptor is explained by a global linear direction in the tangent-space representation; values are averaged over 10 random seeds in Table S2.Source: Table S2 · p. 22, Figure S1 · p. 23

Related papers

Loading 2609.09573v1…