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Estimation of perceptual scales using ordinal embedding

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Pith's one-line read Ordinal embedding from machine learning, applied to triplet comparisons, estimates perceptual scaling functions as accurately as MLDS in the standard one-dimensional monotonic case and also handles non-monotonic and multi-dimensional…

desk verdict Solid methods-transfer paper: the 1D monotonic and non-monotonic results hold up, but the multi-dimensional 'desirable scaling function' claim is under-validated and should be tempered or better anchored. read the letter →

arxiv 1908.07962 v1 pith:UPTZR5JX submitted 2019-08-21 cs.LG cs.CVstat.ML

Siavash Haghiri , Felix Wichmann , Ulrike von Luxburg This is my paper · ORCID
classification cs.LGcs.CVstat.ML
keywords perceptualscalingordinalembeddingtripletcomparisonsmethodoftriadsmaximumlikelihooddifferencestochasticpsychophysicsmulti-dimensional
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes that ordinal embedding methods from machine learning—especially t-STE—can serve as a general tool for estimating psychophysical scaling functions from triplet comparisons of the form 'is stimulus i more similar to j or to k?'. Across simulations and two real psychophysics experiments, the methods match the accuracy of maximum likelihood difference scaling (MLDS) when the perceptual scale is one-dimensional and monotonic. They also recover non-monotonic scaling functions and multi-dimensional perceptual spaces, cases MLDS cannot handle by construction, and they need only on the order of $d n \log n$ triplet judgments instead of the full dissimilarity ordering required by NMDS. If the paper is right, psychophysics gains a default scaling algorithm that keeps the strengths of MLDS while dropping its monotonicity and one-dimensionality restrictions.

What carries the argument

The central object is the ordinal embedding problem: given triplet answers, find points $y_1,\dots,y_n$ in $d$-dimensional Euclidean space whose distances are consistent with the answers. The workhorse is stochastic triplet embedding (STE), which models the probability that stimulus $i$ is judged closer to $j$ than to $k$ as $p_{ijk} = \frac{\exp(-\lVert y_i - y_j \rVert^2)}{\exp(-\lVert y_i - y_j \rVert^2) + \exp(-\lVert y_i - y_k \rVert^2)}$, then maximizes the likelihood of the observed answers; the t-STE variant replaces the Gaussian kernel with a heavy-tailed Student-t kernel for robustness to noise. This machinery carries the argument because it imposes neither monotonicity nor a one-dimensional target, and it is paired with the cross-validated triplet error as a ground-truth-free way to judge whether a recovered scale is any good.

What would settle it

Conduct a triplet experiment on a stimulus set known to violate the Euclidean metric (e.g., similarity judgments that fail the triangle inequality); if t-STE's cross-validated triplet error remains far above the repeat-answer baseline while a non-Euclidean model fits well, the Euclidean assumption is the load-bearing weakness.

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Extended reading notes

Core claim

The discovery is that the classic scaling problem can be re-cast as ordinal embedding: treat the stimuli as abstract items, the triplet answers as ordinal constraints, and the recovered Euclidean coordinates as the perceptual scale. The paper shows that stochastic triplet embedding, in particular the t-distributed variant t-STE, produces scaling functions whose cross-validated triplet error is on par with MLDS in the one-dimensional monotonic setting, while MLDS fails on non-monotonic functions and cannot produce multi-dimensional scales at all. In a simulation built from color-similarity data, the embedding methods reconstruct the two-dimensional color circle from a fraction of the triplets, where NMDS loses the structure to noise. In a real slant-from-texture experiment with eight observers, the ordinal embedding methods match MLDS's triplet error and, unlike MLDS, reveal non-monotonic perceptual patterns in two observers. The paper presents this as evidence that ordinal embedding methods are promising default scaling algorithms.

Load-bearing premise

The approach assumes perceived dissimilarity is Euclidean and low-dimensional, so if the true perceptual geometry is not Euclidean the recovered embedding can be a best-fit artifact rather than the actual perception.

Editorial extensions

If this is right

  • If the claim holds, t-STE can serve as a default scaling method in the standard one-dimensional monotonic setting, matching MLDS accuracy when enough triplets are collected.
  • Non-monotonic perceptual functions such as U-shaped or sinusoidal scales become estimable from triplet data, a case where MLDS systematically produces the wrong function.
  • Multi-dimensional perceptual spaces such as color or pitch become recoverable from triplet judgments, extending scaling to settings MLDS cannot address.
  • Because about $d\,n\log n$ randomly sampled triplets suffice, experiments are feasible where collecting a full dissimilarity ordering would be impossible.
  • The cross-validated triplet error supplies a ground-truth-free criterion for judging when an embedding is good enough, usable for MLDS and NMDS as well.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial: if the Euclidean model is wrong, a t-STE embedding could fit triplet answers well while misrepresenting the true geometry, so testing on known non-Euclidean similarity spaces would delimit when the method is trustworthy.
  • Editorial: because embeddings are only defined up to rotation, reflection, and scaling, comparing observers requires an alignment step; adding such an alignment could make the method a tool for studying inter-observer agreement in perceptual geometry.
  • Editorial: the $d\,n\log n$ sample bound invites adaptive triplet selection that targets comparisons near estimated equality boundaries, potentially reducing data requirements below random sampling.
  • Editorial: the same triplet-embedding formulation should transfer to conjoint measurement and multimodal stimuli, offering a way to map combined perceptual spaces without additivity or independence assumptions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proposes using ordinal embedding methods (STE, t-STE, LOE) to estimate perceptual scaling functions from triplet comparison data, in contrast to the traditional psychophysical methods NMDS and MLDS. It argues that ordinal embedding requires fewer triplets than NMDS, does not impose monotonicity as MLDS does, and can handle multi-dimensional perceptual spaces. The authors support this with simulations covering one-dimensional monotonic and non-monotonic scaling functions and a two-dimensional perceptual space, plus two real experiments (slant-from-texture and Eidolon image distortion). They conclude that in the one-dimensional monotonic case ordinal embedding performs comparably to MLDS, while in non-monotonic and multi-dimensional cases it is the only method among those considered that yields a desirable scaling function.

Significance. If the central claim holds, the paper provides psychophysics with a practical 'default' scaling algorithm based on triplet comparisons, with weaker assumptions than MLDS and lower data requirements than NMDS. The simulation study is extensive and well controlled, covering four noise levels and five triplet fractions, and the real experiments use cross-validated triplet error with a useful human-baseline calibration for the Eidolon data. The comparison with MLDS in the monotonic one-dimensional case is informative, and the non-monotonic one-dimensional simulation makes a clear point. The main limitation is that the multi-dimensional part of the central claim is only validated by a simulation whose ground truth is itself an NMDS Euclidean embedding, and by a real experiment that checks predictive consistency but not the interpretability of the recovered dimensions.

major comments (2)
  1. [Section 3.3, Figure 6(a)] The only multi-dimensional simulation uses as ground truth the two-dimensional NMDS embedding of Ekman's color-similarity data, with the authors explicitly stating that 'we do not argue that this embedding is "correct" in any way; we just use it as a ground truth.' Because NMDS and the tested ordinal-embedding methods both model dissimilarity with Euclidean distances, this simulation demonstrates that t-STE can recover an NMDS-style Euclidean configuration from triplet answers, but it does not validate that the recovered coordinates equal the true perceptual scale. The abstract's claim that in higher dimensions 'only our ordinal embedding methods can produce a desirable scaling function' therefore rests on this simulation plus the Eidolon experiment, and the latter provides only predictive consistency, not perceptual interpretability.
  2. [Section 4.1, Figure 9] The evidence that the Eidolon perceptual space is two-dimensional is the decrease in cross-validated triplet error from d=1 to d=2. Cross-validated triplet error measures agreement with the same type of triplet answers used for training; an embedding that fits triplets well need not have coordinate axes that correspond to any meaningful perceptual attribute. The paper itself lists 'Interpreting the embedding' as an open issue in Section 6.1, and no analysis is reported relating the recovered coordinates to reach, grain, or coherence. To support the multi-dimensional scaling claim, the authors should either provide an external validation (for example, correlation of embedding coordinates with the generation parameters, or a separate judgment task) or explicitly limit the claim to predictive consistency of triplet answers.
minor comments (5)
  1. [Section 3.2.1] The text says the average MSE and triplet errors for the monotonic simulation are depicted in 'Figure 5 (d) and (e)', but the relevant panels are in Figure 4; this cross-reference should be corrected.
  2. [Figure 6 caption] The caption describes the comparison as 'the ordinal embedding methods (MLDS, STE and TSTE)' but MLDS is not an ordinal embedding method; the caption should list LOE, STE, and t-STE.
  3. [Section 2.3.2] There is a typo in the text: 'called chroma and height by Shapard' should read 'by Shepard'.
  4. [Section 4.1] The side experiment on triplet difficulty should clarify the relationship between the measured percentage of inconsistent repeated answers (9.2%, 9.8%, 11%) and the statement that 'we would expect at least 0.10 triplet error'; as written, the step from the observed inconsistency rate to the 0.10 error floor is not immediate.
  5. [Section 3.1] The description of choosing the best of 10 restarts by 'the least triplet error' should specify whether this is the training-set triplet error, since the definition in Equation (5) allows both training and validation variants.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: evaluation uses held-out triplet error and ground-truth MSE, with no fitted parameter or self-citation chain forcing the central claims.

full rationale

The derivation chain is self-contained against independent benchmarks. In simulations, the authors generate triplet answers from explicitly specified ground-truth scaling functions and evaluate with MSE against those functions: “we can compute the mean-squared-error (MSE) between the estimated scales ŷ and the true perceptual function values y” (Section 3.1), so the quality measure is not the training objective. For real data, the paper consistently uses cross-validated triplet error and warns that the training-set version “can be highly biased and typically underestimates the true triplet error (overfitting)” (Section 5.3), so no training-set prediction is relabeled as a result. The only multi-dimensional simulation uses an NMDS embedding of Ekman's color data as ground truth and explicitly disclaims it: “we do not argue that this embedding is ‘correct’ in any way; we just use it as a ground truth to generate further simulations” (Section 3.3). That is a same-model benchmark (both NMDS and ordinal embedding are Euclidean), but it is an honest simulation limitation, not a circular derivation, because the NMDS configuration is not an input to t-STE and the triplets are generated with added noise. Likewise, the Eidolon experiment is evaluated on held-out triplet prediction, and the paper itself flags the absence of an external anchor: “Interpreting the embedding ... What does each perceptual dimension mean? How are the perceptual dimensions related to the parameters of the stimulus (in this case reach, coherence and grain)?” (Section 6.1). Self-citations (LOE, uncertainty estimates, comparison-based nearest neighbors) are background or baseline algorithms and are not load-bearing; no uniqueness theorem is invoked to forbid alternatives, and t-STE is taken from the external implementation of Van Der Maaten and Weinberger (2012) with fixed default parameters. The central 1D monotonic claim is additionally checked against MLDS with the same triplet inputs and against NMDS, so the paper's main comparisons do not reduce to its own definitions.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the Euclidean representability of perceptual space, a probabilistic response model for triplet answers, and the choice of embedding dimension in real experiments. No new entities are postulated, and no parameters are fitted to the real data beyond the standard algorithm hyperparameters.

free parameters (1)
  • embedding dimension d = 1 to 8 in Eidolon; set to true dimension in simulations
    The chosen dimension determines the recovered perceptual space; in real experiments it is selected by cross-validated triplet error.
assumptions (3)
  • domain assumption Perceptual dissimilarity can be represented by Euclidean distances in a low-dimensional space.
    Section 2.3.1 defines the ordinal embedding goal as finding Euclidean points consistent with triplet answers; this assumes the perceptual space is Euclidean.
  • domain assumption Triplet responses follow a probabilistic model based on embedding distances (STE) or are generated by a noisy observer model.
    Section 2.3.3 introduces Equation (4) for STE; Section 3.1 uses a Gaussian noise model for simulated observers.
  • domain assumption Observers answer triplet questions independently.
    Used in the likelihood formulation in Section 2.3.3.

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Cite this review

Pith. "Pith review of Estimation of perceptual scales using ordinal embedding." pith.science (2026). https://pith.science/paper/UPTZR5JX

@misc{pith2026190807962,
  author       = {Pith},
  title        = {Pith review of: Estimation of perceptual scales using ordinal embedding},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UPTZR5JX}},
  note         = {Machine review of arXiv:1908.07962}
}
read the original abstract

In this paper, we address the problem of measuring and analysing sensation, the subjective magnitude of one's experience. We do this in the context of the method of triads: the sensation of the stimulus is evaluated via relative judgments of the form: "Is stimulus S_i more similar to stimulus S_j or to stimulus S_k?". We propose to use ordinal embedding methods from machine learning to estimate the scaling function from the relative judgments. We review two relevant and well-known methods in psychophysics which are partially applicable in our setting: non-metric multi-dimensional scaling (NMDS) and the method of maximum likelihood difference scaling (MLDS). We perform an extensive set of simulations, considering various scaling functions, to demonstrate the performance of the ordinal embedding methods. We show that in contrast to existing approaches our ordinal embedding approach allows, first, to obtain reasonable scaling function from comparatively few relative judgments, second, the estimation of non-monotonous scaling functions, and, third, multi-dimensional perceptual scales. In addition to the simulations, we analyse data from two real psychophysics experiments using ordinal embedding methods. Our results show that in the one-dimensional, monotonically increasing perceptual scale our ordinal embedding approach works as well as MLDS, while in higher dimensions, only our ordinal embedding methods can produce a desirable scaling function. To make our methods widely accessible, we provide an R-implementation and general rules of thumb on how to use ordinal embedding in the context of psychophysics.

Figures

Figures reproduced from arXiv: 1908.07962 by the authors.

Figure 1
Figure 1. An example of a scaling function. The X-axis shows the physical stimulus values ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Top: Eight stimuli used in the slant-from-texture experiment (Aguilar et al., 2017). Bottom, left: [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Left : The two-dimensional circle of color perception gathered by similarity measurements between [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Comparison of various ordinal embedding methods (LOE, STE, t-STE) against the traditional [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Comparison of various ordinal embedding methods (LOE, STE, t-STE) against the traditional [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Comparison of the ordinal embedding methods (MLDS, STE and TSTE) against the traditional [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: (Top) Average and standard deviation of cross-validated triplet error for 8 subjects of the slant-from [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Left: The original image from our Eidolon experiment. Right: An example triplet question — [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Cross-validated triplet error of three embedding methods for three subjects of the Eidolon experiment. [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: The comparison of various ordinal embedding methods (LOE, STE, t-STE) against the traditional [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: The comparison of various ordinal embedding methods (LOE, STE, t-STE) against the traditional [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: The comparison of various ordinal embedding methods (LOE, STE, t-STE) against the traditional [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: The comparison of various ordinal embedding methods (LOE, STE, t-STE) against the traditional [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
Figure 14
Figure 14. Figure 14: The comparison of various ordinal embedding methods (LOE, STE, t-STE) against the traditional [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]

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    2017

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.