Geometric organization of olfactory descriptor data in the Poincaré disk

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

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Geometric organization of olfactory descriptor data in the Poincaré disk

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

arXiv:2609.09573v1Sep 9, 2026cs.CEcs.LGstat.AP

TL;DR

Ol odor descriptor data lack an established low-dimensional coordinate system, limiting interpretable organization of odor quality. This study uses two-dimensional hyperbolic embedding to reveal complementary radial and angular structure across continuous ratings and binary annotations.

Problem
Olfaction lacks an established, widely accepted low-dimensional mapping comparable to coordinate systems for visual and auditory perception.
Method
The study applies hyperbolic metric multidimensional scaling to continuous Sagar ratings and binary GoodScents–Leffingwell annotations, analyzing radial and angular organization.
Results
The embeddings revealed complementary organization: profile entropy varied radially, while descriptor gradients and related binary descriptors formed coherent angular regions.
Takeaways & Limitations
Hyperbolic mapping provides an interpretable descriptive framework in which radius summarizes global profile properties and angle captures descriptor-specific and categorical structure.
Takeaways & Limitations
The study uses descriptor data and does not directly test neural mechanisms of olfactory coding.

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.

ORCID iDs

The section lists the ORCID iDs of four authors.

ORCID iDs: ORCID iDs are provided for Aniss Aiman Medbouhi, Farzaneh Taleb, Giovanni Luca Marchetti, and Danica Kragic.The identifiers are 000-0002-6649-3325, 0000-0003-4482-1460, 0009-0004-8248-229X, and 0000-0003-2965-2953, respectively.

Correspondence to be sent to

The paper concerns olfactory perception, hyperbolic geometry, and dimensionality reduction.

The study focuses on olfactory perception.
It applies hyperbolic geometry as a conceptual framework.
Dimensionality reduction is another central topic.

1 Introduction

Because olfactory perception lacks an agreed low-dimensional coordinate system and depends on context, vocabulary, and individual differences, this study evaluates two-dimensional hyperbolic mapping as an interpretable descriptive framework. Using complementary continuous-rating and binary-annotation datasets, it identifies complementary radial and angular organization of odor descriptor profiles.

Motivation: The study addresses the lack of an agreed dimensional organization for odor quality by representing similarities and differences among odor percepts or descriptor profiles rather than universal molecular-to-experience mappings.Existing odor maps depend on experimental setup, vocabulary, concentration, task, and individual or cultural perceptual and verbal strategies.
Methodological framing: The two-dimensional hyperbolic representation is chosen for visualization and interpretation, distinguishing center-to-boundary variation from angular organization without claiming intrinsic two-dimensional or hyperbolic olfactory representations.Hyperbolic space provides exponentially increasing available area with distance from the origin, a property useful for branching or hierarchical organization.
Datasets: Two complementary datasets enable analysis of graded perceptual profiles and large-scale binary descriptor organization.Sagar contains continuous ratings from three participants for monomolecular odorants, while GoodScents–Leffingwell contains binary expert annotations for several thousand molecules.
Study contribution: The present study extends preliminary Sagar analyses into a broader statistically validated framework across both datasets.Earlier conference-abstract analyses used hyperbolic contrastive learning and identified an initial association between descriptor-profile entropy and hyperbolic radius.
Study contribution: Hyperbolic mapping separates global descriptor-profile organization from descriptor-specific gradients and categorical odor-quality structure, revealing complementary radial and angular organization.In Sagar, diffuse profiles lie nearer the center and concentrated profiles nearer the boundary; in GoodScents–Leffingwell, related binary descriptors occupy coherent disk regions, with entropy relationships depending on definition.

2 Materials and methods

The study constructs common descriptor representations, learns two-dimensional Poincaré-disk embeddings with hyperbolic metric multidimensional scaling, and evaluates both distance preservation and interpretable radial and angular organization. It analyzes continuous subject-specific ratings and binary expert annotations using entropy, tangent-space regression, and descriptor-density structure.

Evaluation framework: Two-dimensional hyperbolic embeddings are evaluated first for pairwise distance preservation, then for radial and angular organization of entropy, ratings, and binary labels.The embedding is treated as interpretable only after verifying that it retains the input descriptor geometry.
Datasets: The Sagar dataset contains 480 subject–odorant observations from three subjects, represented in a shared 15-dimensional space of normalized continuous perceptual ratings.Subject-specific descriptors are excluded to ensure a common descriptor space across participants.
Datasets: The GoodScents–Leffingwell dataset contains 4983 molecules, each represented by a 138-dimensional binary vector of expert-defined odor descriptors.Each molecule appears once, providing a large-scale binary annotation dataset without subject-level repeated measures.
Poincaré model: Hyperbolic metric multidimensional scaling learns embeddings by preserving pairwise input distances in the Poincaré disk, where distances expand toward the boundary.Coordinates are optimized with Riemannian Adam after initialization in the disk.
Entropy analysis: Entropy summarizes descriptor organization, including robustness analyses based on orthogonalized modes of variation rather than individual descriptor coordinates.The same entropy formula is used for both datasets, while probability-vector construction differs by data type.
Radial and angular analyses: Directional variation is analyzed in the tangent space at the disk origin, enabling linear regression while retaining radial and angular organization; binary labels are assessed for coherent high-density regions.Angular orientation is not intrinsically fixed because embeddings may rotate or reflect across initializations.

3 Results

Hyperbolic embeddings preserved olfactory descriptor geometry while revealing complementary radial and angular organization. In Sagar, radius tracked profile entropy and descriptor directions, whereas in GoodScents–Leffingwell, active-label entropy increased with radius.

Radial organization: Sagar embeddings preserved pairwise descriptor geometry, with distance Pearson and Spearman correlations of 0.82±0.02 and 0.81±0.02.These embedding-level metrics were shared by the entropy and intensity analyses.
Radial entropy organization: Sagar profile entropy correlated negatively with radius, with Pearson −0.77±0.05 and Spearman −0.72±0.05, placing diffuse profiles centrally and concentrated profiles peripherally.The association remained strong across descriptor-removal, pruning, orthogonalization, participant-specific, and averaged-rating analyses.
Radial organization: Intensity was the only individual Sagar descriptor with a radial association, showing Pearson 0.42±0.05 and Spearman 0.38±0.05 correlations.All other descriptor-level radial correlations were below 0.2 in absolute value.
Angular organization: Angular organization was strongest for sweet, musky, fruity, and pleasantness, whose mean directional R2 values ranged from 0.40 to 0.51.These trends were supported by restricted permutation tests with pWS < 0.001 and pOB < 0.001.
GoodScents–Leffingwell: GoodScents–Leffingwell embeddings preserved binary descriptor geometry with distance Pearson 0.69±0.01 and Spearman 0.68±0.01, while active-label entropy rose with radius.Active-label entropy correlated positively with radius at Pearson 0.87±0.03 and Spearman 0.88±0.03.

4 Discussion

The discussion concludes that the Poincaré disk provides an interpretable decomposition of olfactory descriptor data into global radial properties and descriptor-specific or categorical angular organization. It also identifies limitations involving neural validation, sample size, qualitative binary-descriptor analysis, molecular structure, and unmeasured individual-difference factors.

Interpretation: Hyperbolic radius summarized global descriptor-profile properties, while angular structure captured continuous descriptor gradients and localized categorical organization across the Sagar and GSLF datasets.In Sagar, radius was associated with rating profile entropy; in GSLF, active label entropy increased with radius because peripheral molecules had more descriptors.
Directional organization: Sweet, musky, fruity, pleasantness, and decayed were well summarized by dominant tangent-space directions, with pleasantness showing a particularly strong directional trend.The discussion relates pleasantness’s directional organization to prior work identifying it as an important axis of olfactory perception.
Limitations and future work: The study does not directly test neural mechanisms, so linking the observed non-Euclidean descriptor geometry to olfactory representations will require sufficiently large and diverse fMRI or EEG datasets.The present results are compatible with non-Euclidean olfactory structure but do not establish its neural basis.
Limitations and future work: The continuous-rating dataset included few subjects, limiting conclusions about the consistency and population variability of radial and directional organization.Larger, more diverse samples and richer vocabularies from trained assessors are proposed for future characterization.
Limitations and future work: Hyperbolic KDE regions for binary descriptors provide qualitative evidence of coherent label regions but not formal tests of category separation.Future analyses could quantify spatial concentration, overlap, and separation among descriptor families.
Limitations and future work: The analysis omitted molecular structure, behavioral confidence, emotion, cognitive style, personality, and clinical olfactory measures, limiting mechanistic interpretation of geometric and subject-specific differences.Joint molecular-perceptual hyperbolic models and additional individual-difference measures could clarify the origins and correlates of the observed organization.

Supplementary material · A Hyperbolic geometry and optimization details

This supplementary section provides the geometric expressions and optimization details underlying the hyperbolic metric MDS model. It supplies general Poincaré ball expressions and Riemannian optimization updates for reproducibility, while the main text presents the concepts needed for radial and directional analyses.

Supplementary material: The supplementary material documents the geometric expressions underlying the hyperbolic metric MDS model.
Supplementary material: It also specifies the optimization details used by the hyperbolic metric MDS model.
A Hyperbolic geometry and optimization details: General Poincaré ball expressions are provided to support reproducibility.
A Hyperbolic geometry and optimization details: Riemannian optimization updates are provided as part of the reproducible technical specification.
Supplementary material: The main text contains the concepts needed to understand the radial analyses.
Supplementary material: The main text also contains the concepts needed to understand the directional analyses.

A.1 Geometry of the Poincaré ball

The study uses the Poincaré ball as a model of n-dimensional hyperbolic space, with a curvature-dependent Riemannian metric and geodesic distance. Near the boundary, equal Euclidean displacements correspond to increasingly large hyperbolic distances.

Poincaré ball model: The Poincaré ball models n-dimensional hyperbolic space as a simply connected manifold of constant curvature −1.The model is the Euclidean ball equipped with a Riemannian metric.
Riemannian metric: Its Riemannian metric scales the Euclidean inner product by a curvature-dependent factor, defining lengths and norms for tangent vectors.
Geodesic distance: The associated geodesic distance is the length of the shortest smooth path between two points in the Poincaré ball.This distance is defined through curve lengths and has an explicit computational expression.
Boundary geometry: Equal Euclidean displacements produce increasingly large hyperbolic distances as points approach the disk boundary.

A.2 Möbius addition · A.3 Tangent spaces and geometric maps · A.4 Parallel transport

The appendix defines Möbius addition and uses it to formulate tangent-space maps and parallel transport on the Poincaré manifold. These geometric operations support distance-preserving tangent analysis and optimization across changing embedding positions.

A.2 Möbius addition: Möbius addition provides the common formulation for the exponential map, logarithmic map, and parallel transport.The operation is defined on P^n, and its additive inverse is the Euclidean negative.
A.3 Tangent spaces and geometric maps: Each tangent space T_zP^n is vector-space identifiable with R^n, but its inner product varies with z through the Riemannian metric.
A.3 Tangent spaces and geometric maps: The exponential map follows the geodesic from z in tangent direction v, sending v ∈ T_zP^n to a manifold point.
A.3 Tangent spaces and geometric maps: The logarithmic map sends y ∈ P^n to the tangent vector at z directed along the geodesic from z to y.
A.3 Tangent spaces and geometric maps: The tangent vector’s Riemannian norm equals hyperbolic distance, whereas its ordinary Euclidean norm generally differs because the metric is scaled by λ_z.Origin-based versions of these maps are used for tangent-space analysis of continuous descriptor ratings.
A.4 Parallel transport: Parallel transport moves tangent vectors along geodesics while preserving Riemannian geometry, because tangent spaces at distinct points cannot be directly combined.
A.4 Parallel transport: The parallel-transport construction uses the gyrovector formalism’s gyration operator to transfer vectors between points on P^n.
A.4 Parallel transport: During optimization, parallel transport transfers the first-moment estimate between successive embedding positions.

A.5 Embeddings initialization

Embeddings were initialized by sampling tangent-space points from a standard Gaussian and mapping them into the Poincaré disk with the exponential map. Independent random initializations used σ_init = 0.1.

A.5 Embeddings initialization: Embeddings were initialized by pushing standard Gaussian samples from the tangent space at the origin through the exponential map into the open Poincaré disk.This pseudo-hyperbolic Gaussian produces valid interior points for each random initialization.
A.5 Embeddings initialization: Independent random initializations were used with σ_init = 0.1 in all experiments.

A.6 Riemannian Adam optimization

Embedding coordinates were optimized on the Poincaré disk with a Riemannian Adam method using closed-form manifold operations, Adam bias correction, and projection for numerical stability. The implementation used full-batch or mini-batch training depending on dataset size and reached stable loss plateaus after 1000 epochs.

Riemannian Adam optimization: Riemannian Adam updated Poincaré-disk embeddings using closed-form Riemannian gradients, exponential maps, parallel transport, bias correction, and unit-disk projection.Parallel transport preserved first-moment momentum across tangent spaces, while projection maintained numerical stability within the open disk.
Riemannian Adam optimization: The implementation used β1 = 0.9, β2 = 0.999, ε = 10−8, η = 0.1, and εproj = 10−5 for all experiments.Sagar optimization used full batches, whereas GoodScents--Leffingwell used mini-batches of 195 molecules.
Riemannian Adam optimization: Training lasted 1000 epochs in every configuration, with the optimization loss reaching a stable plateau by the end.This plateau was reported as indicating convergence.

B Additional results

Supplementary analyses quantify radial and angular descriptor organization in the Sagar union embedding. Radial correlations, directional R2 values, and fitted tangent-space directions are reported across descriptors and random initializations.

Radial descriptor results: Table S1 reports full radial Pearson and Spearman correlations for each descriptor, with means and standard deviations over 10 random seeds and restricted permutation p-values.The permutation tests include within-subject (pWS) and odorant-block (pOB) schemes.
Angular descriptor results: Table S2 reports full angular descriptor results using directional R2 from global linear directions in the Poincaré disk’s tangent-space representation.Values are summarized over 10 random seeds with within-subject and odorant-block restricted permutation p-values.
Angular descriptor results: Figure S1 visualizes directional organization for all 15 continuous descriptors using descriptor-colored Poincaré disk panels and arrows for increasing-value tangent-space directions.The displayed directional angles and R2 values are specific to random initialization m = 5.