LORE: Jointly Learning The Intrinsic Dimensionality and Relative Similarity Structure from Ordinal Data
Authors: Vivek Anand
Alec Helbling
Mark A. Davenport
Gordon J Berman
Sankaraleengam Alagapan
Christopher John Rozell
Venue: ICLR 2026 Poster (2026)
Keywords: Ordinal Embedding
Intrinsic Dimensionality
Psychophysics
Subjective Perceptual Learning
Relative Similarity Embedding
Primary area: applications to neuroscience & cognitive science
TL;DR: LORE jointly infers both the intrinsic dimensionality and an ordinal embedding from noisy triplet comparisons, enabling compact, interpretable perceptual representations from purely ordinal data.

Abstract: Learning the intrinsic dimensionality of subjective perceptual spaces such as taste, smell, or aesthetics from ordinal data is a challenging problem. We introduce LORE (Low Rank Ordinal Embedding), a scalable framework that jointly learns both the intrinsic dimensionality and an ordinal embedding from noisy triplet comparisons of the form, "Is A more similar to B than C?". Unlike existing methods that require the embedding dimension to be set apriori, LORE regularizes the solution using the nonconvex Schatten-$p$ quasi norm, enabling automatic joint recovery of both the ordinal embedding and its dimensionality. We optimize this joint objective via an iteratively reweighted algorithm and establish convergence guarantees. Extensive experiments on synthetic datasets, simulated perceptual spaces, and real world crowdsourced ordinal judgements show that LORE learns compact, interpretable and highly accurate low dimensional embeddings that recover the latent geometry of subjective percepts. By simultaneously inferring both the intrinsic dimensionality and ordinal embeddings, LORE enables more interpretable and data efficient perceptual modeling in psychophysics and opens new directions for scalable discovery of low dimensional structure from ordinal data in machine learning.

Published as a conference paper at ICLR 2026
LORE: JOINTLY LEARNING THE INTRINSIC DIMENSIONALITY AND RELATIVE SIMILARITY STRUCTURE FROM ORDINAL DATA
Vivek Anand"; Alec Helbling, Mark A. Davenport, Sankaraleengam Alagapan, Christopher John Rozell Georgia Institute of Technology
Atlanta, GA, USA
(vivekanand) daatenddd@gaechheee..
Gordon J. Berman
Emory University Atlanta, GA, USA
ABSTRACT
Learning the intrinsic dimensionality subjective perceptual spaces such as taste, smell, aesthetics from ordinal data is challenging problem. We introduce LORE (Low Rank Ordinal Embedding), a scalable framework that jointly learns both the intrinsic dimensionality and ordinal embedding from noisy triplet com- parisons of the form, "Is A more similar to B than C?". Unlike existing methods that require the embedding dimension be set apriori, LORE regularizes the solution using the nonconvex Schatten-p quasi norm, enabling automatic joint re- covery of both the ordinal embedding and its dimensionality. We optimize this joint objective via an iteratively reweighted algorithm and establish convergence guarantees. Extensive experiments on synthetic datasets, simulated perceptual spaces, and real world crowdsourced ordinal judgements show that LORE learns compact, interpretable and highly accurate low dimensional embeddings that re- cover the latent geometry of subjective percepts. By simultaneously inferring both the intrinsic dimensionality and ordinal embeddings, LORE enables more inter- pretable and data efficient perceptual modeling in psychophysics and opens new directions for scalable discovery low dimensional structure from ordinal data machine learning.
INTRODUCTION
Learning subjective percepts (SPs), such as taste, smell, or aesthetic preference, poses unique chal- lenges for machine learning. Traditional approaches rely on absolute queries that presuppose known perceptual axes. For example, taste study might ask participants to rate stimuli on 1-5 Likert scale (Likert 1932) for "sweetness" or "bitterness". Such methods suffer from two critical flaws: (1) inconsistency, respondents interpret scales differently (c.g., one person's 's "moderately sweet" is another's "very sweet") (Stewart et al.) 2005), and (2) predefined conceptual frameworks that limit discovery by forcing ratings predefined axes. Consequently, researchers risk missing latent dimensions (e.g., a "metallic" undertone in coffee) that participants lack vocabulary to describe.
In contrast, relative queries circumvent these issues by capturing perceptual relationships directly. For example, a triplet comparison like "Is coffee A more similar to coffee B or coffee C in taste?" allows participants express nuanced judgments without relying on language or preset scales. Such relative comparisons are therefore particularly well suited for discovering the latent dimensions that organize subjective perceptual spaces.
Relative Similarity or Ordinal Embedding methods (OE) leverage these relative judgements to learn a multidimensional representation. However, all existing OE approaches require the user to specify
"Corresoondigg Author

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Published as a conference paper at ICLR 2026
Overparameterized
Perceptual Representation
Other Methods susceptible to incorrectly specified embedding dimension
Underfits Data
LORE jointly learns the relative similarities and the intrinsic climensionality
Recovers Parsimonious
Perceptual Space
Figure 1: LORE jointly learns both the intrinsic dimensionality and relative similarities by balancing dimensionality with similarity constraints: Other methods are susceptible to an incor- rectly chosen embedding dimension.
the embedding dimension in advance Terada & Luxburg 2014 Agarwal et al. 2007 Tamuz et al, 2011 Jain al., 2016, Van Der Maaten & Weinberger 2012), with little guidance to the "true" complexity of the perceptual space. In practice, this can lead to unnecessarily high dimensional embeddings, concealing the actual structure. For instance, an OE may perfectly satisfy all triplet constraints in a 10-dimensional space, even if the underlying percept is only 2-dimensional.
Scientific discovery demands parsimony, a principle formalized Occam's razor (Bishop & Nasrabadi 2006). For the taste example, 2D embedding is preferable to a 10D alternative: it casier to interpret, less computationally intensive, and more useful for downstream analyses. In practical terms, 10D taste embedding might fragment "sweetness" into several axes, complicating flavor design or neurological interpretation. Yet, most OE approaches, despite high triplet accuracy, produce overly complex models that mask the true structure of the latent percept.
To address this gap, We introduce LORE, a new ordinal embedding algorithm that jointly learns both the embedding and the intrinsic dimensionality, instead of needing to specify the dimension apriori. LORE regularizes using the nonconvex Schatten-p quasi-norm, explicitly balancing triplet accuracy with representation compactness and is optimized via an iteratively reweighted algorithm, with guarantees of convergence to stationary points. Our main contributions are:
1. LORE, novel ordinal embedding algorithm that recovers latent representations that match the intrinsic dimensionality of human perceptual similarity data. LORE jointly infers both the embedding and its dimensionality by regularizing with the noncon- vex Schatten-p quasi-norm. By balancing triplet accuracy and rank regularization we can infer compact yet accurate representation avoiding underfitting overparameterization. We optimize the resulting objective using an iteratively reweighted schattent quasi norm algorithm, and provide convergence guarantees of the OE stationary points.
2. LORE reliably uncovers the intrinsic dimensionality of data through an extensive evaluation where the dimensionality is known apriori. We first extensively test our algorithm on data with various dimensionality, noise levels and number of queries and demonstrate it outperforms existing methods by far in estimating intrinsic dimensionality with close to optimal performance in triplet accuracy. Secondly, we conduct a simulated perceptual experiment to model taste using LLM the ground truth perceptual space we try to model. See Figure for high level summary of our results.
3. LORE outperforms numerous state of the art methods on the large crowd sourced datasets and learns semantically interpretable axes. We find that LORE achieves a lower rank representation compared all baselines while achieving comparable triplet accuracy on three separate crowdsourced datasets (Wilber et al.) 2014 Kleindessner & Von Luxburg, 2017 Ellis et al. 2002) and learss axes which are semantically interpretable.
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We anticipate LORE will be valuable tool for mapping subtle subjective phenomena to inter- pretable oow-dimensional spaces across psychology, neuroscience, and social science. By removing the need to hand tune embedding dimension, LORE jointly learns both the relative similarities and recover true intrinsic dimensionality enabling data driven discovery of subjective percepts.
2 RELATED WORK
Ordinal Embeddings as Tools for Psychophysical Scaling: Psychophysics aims to discover the quantitative mappings that humans use to connect external stimuli to inner perceptual experiences. Psychological percepts are usually studied via relative judgements as they are less prone to indi- vidual biases, scale interpretation and memory limitations than absolute judgements as humans do not perceive stimuli in isolation (Stewart et al. 2005). Given the constraints of data collection, core challenge in psychophysics is reconstructing perceptual spaces from few human similarity judgments. OEs address this challenge; they are both query efficient and capable of reconstructing multidimensional perceptual spaces. For example, Filip et al.) (2024) derived a tactile-visual em- bedding for wood textures, identifying roughness and gloss as perceptually orthogonal dimensions. Huber et al. (2024) used one such OE to map philosophical concepts onto conceptual axes from human similarity data and [Sauer et al. (2024) used it map perceived distortions of vision from spectacles. Moreover, active learning approaches like Canal et al. (2020) have demonstrated how query efficiency for data collection can be further improved.
Learning are distinct from OEs: Learning from relative compar- isons has been used in metric learning "Suárez-Diaz et al., 2018 and contrastive learning (Chen et al. 2020). Metric learning aims to learn a metric space from the data while contrastive learning separates similar and dissimilar datapoints. Metric Learning typically combines relative judgements and explicit representations (say images). The goal is to learn a distance metric from both sources of information. This additional representation, absent in OEs, changes the optimization problem and prevents direct transfer of metric learning approaches. Contrastive learning seeks to group similar datapoints together and push dissimilar ones apart with the presence of explicit additional supervised supiiion information which do not exist for OFs. Therefore, while metric learning and contrastive learning methods are similar in learning from relative information, they cannot be directly applied to OEs.
Metric
Intrinsic Dimensionality recovery is critical for psychophysics: A core goal in psychophysics is to recover the latent internal representations that individuals use to perceive psychophysical stim- uli. Each representation is composed of two important characteristics: how well the representation recovers the ordinal relationships between the percepts and the intrinsic runk or dimensionality of the representation obtained. While OEs are able maintain ordinal consistency (Vankadara al. 2023). they are unable to identify the intrinsic dimensionality as we show in this paper. This is a key limitation of OFs that reduces their utility for psychophysical analysis. Künstle et al. (2022) ad- dressed this problem by modelling it as multiple hypothesis test with separate embeddings trained for each candidate dimension and triplet accuracies used estimate the true intrinsic rank. This approach, however, has two main limitations:
Hypothesis dependence: It requires predefining plausible dimensionalities, risking model misspecification and reducing statistical power if the true dimensionality exceeds or is less than the hypothesized bounds.
2. Lack of Scalability: Training multiple embeddings for each hypothesized rank is com- putationally expensive and quickly becomes prohibitive for a greater number of percepts. This is especially problematic for active querying where efficiency critically important.
Building on these limitations, we propose method to jointly infer both dimensionality and mul- tidimensional representations via a novel OE method, eliminating the need for explicit hypothesis enumeration. For psychophysics, this enables recovery of perceptual geometry without prior sumptions on dimensionality. For machine learning, it offers a scalable approach to uncovering low dimensional structure directly from ordinal data.
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Table Characterization of Different Ordinal Embedding Algorithms
Optimizes
Recovers Intrinsic Rank
High Triplet Accuracy
Semantically Interpretable
Method
Scalable
Over
GNMDS
Gram Matrix
CKL
Gram Matrix
x
FORTE
Gram Matrix
X
x
t-STE
Embedding
X
SOE
Embedding
X
OENN
Embedding
X
X
LORE (ours)
Embedding
BACKGROUND ON ORDINAL EMBEDDINGS
The ordinal embedding problem seeks to learn an embedding matrix Z E Nxxd from triplet judge- ments from the true perceptual space lying in unknown P E RNxx where < N is the intrinsic dimensionality or the intrinsic rank of the perceptual space. OE problems usually assume integral dimensionalities/rankss due low number of percepts and do the same. Z is learned indirectly via noisy similarity triplet comparisons where the anchor percept a is more similar or closer in the perceptual space to percept i than percept j into an embedding space of dimension d'. Specifically, this is denoted by (a, j.
E T where d(Pa.; a.:; P.:) < d(Pa,. a, P Pj:)) where P P... Pj. are the rows indexed by percepts a, i.j respectively in and -) the Euclidean distance between the unknown percepts. A central challenge is that intrinsic rank is unknown and the embedding dimension is set heuristically.
Though this framework is relatively simple, solving OEs efficiently can be challenging as the OE problem is NP-Hard (Bower et al., 2018), most loss functions are nonconvex and efficient learning demands at least ( d log N) actively sampled triplets (Jain et al. 20016). As result, the choice of optimization framework is crucial to obtaining a good OE and depending on dataset characteristics, different OE methods may be preferred for different situations (Vankadara et al. 2023).
Gram matrix approaches optimize a positive semidefinite matrix G = ZZT E N that capture the pairwise differences. While theoretically appealing because they agnostic to the embed- ding dimension during optimization, they require enforcing PSD constraints that are not scalable for large N. Early methods like Generalized Non Metric Multi Dimensional Scaling (GNMDS) (Agarwal et al., 2007) and probabilistic models like Crowd Kernel Learning (CKL) suffer from lim- ited accuracy or poor scalability. Fast Ordinal Triplet Embedding (FORTE) accelerates this with kernelized nonconvex triplet loss optimized by efficient Projected Gradient Descent (PGD) and line
search.
Direct embedding approaches optimize Z which leads to faster gradient updates that scale with smaller O ((Nd') versus (N with Gram matrix approaches. Examples include t-distributed Stochastic Triplet Embedding (t-STE) (Van Der Maaten & Weinberger, 2012) and Soft Ordinal Embedding (SOH) (Terada & Luxburg. 2014) with the latter widely used for efficiency and high accuracy. A deep learning variant, Ordinal Embedding Neural Network (OENN) (Vankadara al.
2023) underperforms likely due to the limited supervisory signal in purely ordinal data.
However, shared fundamental limitation of all existing methods is the inability to recover the intrinsic rank d, which risks overparameterizing the true perceptual latent space.
METHODS
We introduce a scalable ordinal embedding (OE) framework that jointly learns both the embedding and the intrinsic rank of the perceptual space. To ensure computational efficiency large datasets (large T and N) we directly optimize the embedding Z instead of the Gram matrix G. The key insight is that
adaptively select the embedding dimensionality as needed to fit the percepts well but not use any more extra space than necessary. Therefore, a natural approach is penalize the rank of the learned embedding via regularization.
want the learning algorithm
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As SOE has the best properties of all the OEs that optimize Z, extend it with regularization. As the rank constraint is NP-Hard and non-convex (Fazel et al. 2001) a common approach is to regularize with the nuclear norm instead where |Z]|. min o, (Z), where (Z) is the i-th singular value (Candes & Recht 2008 Fazel et al. 2001). The objective then becomes:
min 2
(Z)
Σ (a.i.))E
max 1 + d(Zn,; Zea) - d(Za,: 25.)} + A/
(I)
Though the nuclear norm is convex and relatively easy optimize. it uniformly shrinks all of singular values (Negahban & Wainwright 2011 Zhang 2010). Recent theoretical and empiri-
cal evidence indicates that the nonconvex Schatten-p quasi norm | Di=1 [inn o2 (Z))
d. g(a,(Z)) for 0 <
1, recovers the intrinsic rank for low rank recovery problems better than the nuclear norm can (Lu et al. 2014, Marjanovic & Solo 2012). The Schatten-p quasi norm generalizes the nuclear norm by penalizing larger singular values less severely which is shown to aid in intrinsic rank recovery. We leverage this property and for the first time, to our knowledge, integrate the Schatten quasi-norm into a scalable ordinal embedding framework, allowing implicit perceptual rank discovery as seen below in:
Li=1
min Z
(Z)
Σ (a.T
max (0, 1 d(Za.; Z,.) - d(Z 25,)3 + A
Though incorporating the Schatten Quasi-Norm improves rank recovery properties, it also intro- duces additional nonconvexity into the regularizer that makes optimization more challenging. To overcome the inherent con-differentiability of the ordinal loss and the complexity of nonconvex reg- ularization, we smooth the hinge triplet loss with the softplus function (Dugas et al. 2001). This transformation makes the objective differentiable except where the embedding collapses (Zu: Z,,; or Z. Zj.). However, collapses can be avoided with wide initializations of Z. This smoothing enables provable convergence and is empirically essential, as it mitigates zero gradient plateaus to facilitate training large datasets. Then the objective function is defined as:
min [N,d)
min (Z)
Σ (a.i.))T
log(1 + exp(1 + d(Zn... Z..) - d(Za,; Zj.))) +
Σ
(Z). (3)
Despite smoothing the ordinal loss, our objective remains highly nonconvex due to the Schatten-p quasi-norm regularization, which makes reliable optimization difficult. Standard gradient methods often get stuck in poor local minima or fail to converge. To overcome this, we use an iteratively reweighted algorithm inspired by(Sun et al. (2017). At each step, the algorithm minimizes weighted surrogate of the original objective, leading to steady improvement even in complex landscapes. As established in Theorem 1, this procedure is guaranteed converge stationary point, ensuring robust and reliable learning.
Theorem (LORE converges to stationary point). The sequence of OEs generated by the LORE algorithm {Z km 1,2,3... converges. i.e.
2
< +00
(4)
Proof Sketch: We use the general framework for nonconvex Schatten Quasi-Norm optimization as seen /un et al. (2017) ( but crucially, check the specific conditions for the LORE objective. The full proof in Appendix A
Our convergence guarantee is significant because, for ordinal embedding problems, stationary points widely believed to be nearly as good global optima in objective value. This is supported empirically (Vankadara et al. 2023) and theoretically. [ower et al. (2018) proved that for certain OE settings with d 2, all local optima are global. Moreover, when sufficient triplet data is available, sub-optimal local minima are rarely observed. Building on these insights, we expect that method will also recover high quality embeddings in realistic settings. Our experimental results
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Algorithm Learning LORE
1: Input: Z E Nxd' T,
2: Assign prev_
[oo]
3: for k = 0, 1, 2, 2.... do
4: a <- Singular Values (
5: curr_obj < (eT log(1 + exp(1 + d(z,, 24) d(z 2R))) + X > i=
min min(N.dd
(Z)P)
6: if Jeurr_obj prev_objs|__]) tol then
7: break
{Convergence check)
8: end if
9:
U.S. + SVD(Z* /z f(z))
S S Lop-1
11:
S sorted(S* s 0]. descending)
12:
Z t VS
13:
prev objs[k] &- obj
14:
if IZ 1 < tol then
15: break
(Check if close stationary point)
16: end if'
end for
18:
return 1
confirm that LORE learns high accuracy ordinal embeddings, even with the inherent nonconvexity of the objective.
We implement the optimization using an efficient iteratively reweighted algorithm, seen Algo- rithm 1, that updates the embedding and regularization at each step. In the typical regime where the embedding dimension is much smaller than the number of items and triplets, each iteration requires (T N)d)) operations, making LORE scalable large datasets. Additional implementation specifics are in Appendix B
In summary, methodological contributions are: (1) formulating new ordinal embedding ap- proach that jointly learns the ordinal embedding and intrinsic rank using Schatten quasi-norm reg- ularization; (2) establishing efficient optimization strategy based on iteratively reweighted mini- mization tailored for this nonconvex objective, along with convergence guarantees and (3) providing scalable algorithm suitable for large scale perceptual similarity data.
RESULTS
We present five pieces of empirical evidence in support of our method's claims. We outline experimental setup, including baseline methods and the generative process for producing ordinal embedding (OE) tasks. Next, we show how to select regularization levels for LORE. We then benchmark LORE against standard baselines across key metrics, followed by a comparison on proxy large language model (LLM) generated perceptual spaces. Finally, we assess performance on real, crowdsourced triplet data involving human judgments and see that LORE's learned axes have se- mantic meaning. Collectively, these results demonstrate that LORE is the only OE method that can effectively jointly learn high quality ordinal embeddings and the intrinsic rank.
5.1 SETUP
We benchmark LORE primarily against (1) SOE and (2) FORTE. These methods represent the best performing direct and Gram matrix OE approaches, respectively, as established by prior work (Vankadara et al., 2023). For our last two experiments, which are computationally less demanding. also compare against additional OE methods like t-STE, CKL and the dimensionality estimation method using hypothesis testing from (Künstle et al. 2022) denoted as Dim-CV Our core evaluation criteria are:
Test Triplet Accuracy: The proportion of held-out triplets correctly satisfied by the learned embedding and the primary metric in the OF literature.

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154
Test Triplet Accuracy
Fraction of Queries
0.7
10
Rank
0.75
Intrinsic
5
0.3
0.5
10
10
10
10
10
10
100
Figure 2: LORE has high test triplet accuracy and intrinsic rank recovery across varying number of queries. (Left) Mean test triplet accuracy vs À for LORE as Fraction of Queries varies. (Right) Mean measured rank vs À for LORE as Fraction of Queries varies.
Measured Rank:
The effective rank of the learned embedding, as a measure of intrinsic dimensionality recovery.
An ideal ordinal embedding should achieve high test triplet accuracy while maintaining a measured rank close to the true intrinsic rank of the underlying perceptual space.
For LORE, we fix = 0.5 as it offers a good balance between rank recovery and optimization sta- bility seen in prior work (Lu et al. 2014 Wang al. 2024 Sun et al. 2017). Our Results in A [ concur with the prior work. As only needs to be greater than the Lipschitz constant of triplet loss term, set it to 0.1 which is greater than what we find empirically. Both of these are set apriori and do not require tuning. The only hyperparameter that is tuned is the regularization parameter A. However, as we show in Section 5.2 there exists wide range of À around 0.01 that yields both high triplet accuracy and intrinsic rank recovery across a variety of dataset conditions. Thus, we expect that in practice, user can set 0.01 without extensive hyperparameter tuning which our later experiments confirm works well across real world datasets. Initializations are usu- ally critical for nonconvex optimization problems. However, due to the guaranteed convergence of LORE to stationary points, and the fact that Ordinal Embedding algorithms tend to have good local minima we find that random Gaussian initializations with variance of least 5 work well across all experiments without any special tuning. All of our experiments use this initialization scheme and we suggest as a default choice for practitioners.
We systematically vary four factors: fraction of queries, intrinsic rank, number of percepts, and noise level in the generative model. For synthetic experiments, we generate perceptual spaces of specified size and rank, sample noisy triplets mimic human responses, and fit each method before evaluating on test triplets. Our synthetic data model generates a random perceptual space of specified rank and number of percepts, followed by sampling triplets with replacement simulate human queries. We then use a standard approach (Canal et al. 2020; Vankadara et al., (023) to model response uncertainty by sampling Gaussian noise independently and adding to each triplet distance. The resulting triplet data is then used to fit all OE algorithms, which are evaluated on held-out test triplets for both accuracy and measured rank. Unless otherwise stated, all experiments use
query fraction = 0.1, p 0.5, d = 5, N = 50, noise = 0.1, and d' = 15 for 30 independent seeds. We include code to reproduce this generative process in the supplemental material.
5.2 LORE HAS CONSISTENT AND STABLE REGULARIZATION SETTING THAT YIELDS HIGH TRIPLET ACCURACY AND INTRINSIC RANK RECOVERY
Our first set of experiments explores whether LORE admits a regularization regime that yields both high test triplet accuracy and reliable intrinsic rank recovery. As shown in Figure 2 across broad range of the regularization parameter (A 0.01), LORE achieves nearly perfect test triplet accuracy and accurate intrinsic rank recovery, even as the fraction of queried triplets varies. Further results in Appendix C show that LORE performs similarly with varying noise, number of percepts and in- trinsic rank. These pieces of evidence taken together confirm that high triplet accuracy and intrinsic rank recovery persists with different noise levels, numbers of percepts and intrinsic rank for the same regularization range. Thus, LORE is robust in hyperparameter selection (A).

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1.00
15
Text Accuracy
Measured Rank
10
0.75
Rank
Intrinsic Rank
0.50
0.5
0.5
Fraction Queries
Fraction of Queries SOE
Intrinsic Rank
LORE (ours)
FORTE
Figure 3: Only LORE can recover the intrinsic rank while maintaining comparable test triplet accuracy number of queries varies: (Left) Mean test triplet accuracy fraction of queries used. (Center) Mean measured rank vs fraction of queries used. (Right) Mean Measured Rank vs Intrinsic Rank. The gray dotted line indicates the ideal case where the measured rank equal the intrinsic rank. Shaded Areas indicate +2 Standard Deviations.
1.00
10
Test itppet Accuracy
Meas-rrd Bank
10-
0.75
0.50
10
10
10
LORE (ours)
SOE FORTE
CKL
Dim-CV
Figure 4: LORE outperforms baselines for both test triplet accuracy and intrinsic rank for simulated LLM perceptual experiment. (Left) Mean test triplet accuracy intrinsic rank. (Center) Mean measured rank vs intrinsic rank. The gray dotted line is the ideal case where the measured rank equal to the intrinsic rank. (Right) Time taken for processing vs intrinsic rank. Shaded Areas indicate 22 Standard Deviations.
5.3 LORE OUTPERFORMS BASELINES IN RANK RECOVERY; MATCHES IN TRIPLET ACCURACY
Figure (left, center) demonstrates that LORE uniquely recovers the intrinsic rank of the em- bedding across all tested query fractions, while baseline methods consistently default to the maxi- mum allowed dimension. Importantly, LORE matches the test triplet accuracy of the best baseline across all conditions, achieving low rank solutions without sacrificing predictive performance.
Additionally, Figure 3 (right) shows that as the intrinsic rank increases, only LORE tracks this change, whereas all other methods ignore the underlying complexity and fail to adapt. While some loss rank recovery observed higher true ranks (expected due to fixed number of triplets and the curse of dimensionality (Bishop & Nasrabadi] (006)), LORE consistently outperforms competitors recovering reduced the intrinsic rank. Further results Appendix conffim that LORE maintains an advantage across different noise and percept counts. Thus, LORE is the only method to reliably recover both accurate ordinal embeddings and the intrinsic rank thereby avoiding underfitting or overparameterizing the true perceptual space. These results highlight the practical value of LORE for applications where discovering latent structure is critical.
5.4 LORE RECOVERS INTRINSIC RANK IN A SIMULATED PERCEPTUAL EXPERIMENT
Human perceptual experiments are a key application of LORE, but for real perceptual datasets, the true intrinsic rank is unknown. Consequently, leverage recent findings that large language models (LLMs) encode human-alisen -alan- perceptual information across domains such as taste, pitch, and timbre (Marjieh al. 2024). Therefore the LLM embedding space realistic proxy
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of the true perceptual space in humans. To evaluate how our LORE compares to other baseline algorithms and the Dim-CV, which uses hypothesis testing and training multiple embeddings to estimate intrinsic rank, we evaluate both the test triplet accuracy and the measured rank of the learned embeddings. For Dim-CV we keep the range of hypothesed dimensions from 1 to 10 to match the true intrinsic ranks we test here. The measured rank is the averaged resulting rank of the embedding learned by each algorithm across 30 independent runs.
We first obtain SBERT embeddings (Reimers & Gurevych 2019) for 50 randomly chosen foods, then control the intrinsic dimensionality for this experiment by applying truncated singular value decomposition (ranks 1-10). From this lower dimensional space, we generate noisy triplet compar- isons (sampling 5% of the total, 30 runs, noise=0.1) to mimic the data limited regime typical in human perceptual experiments. A detailed experimental setup is included in Appendix ||
Figure 4 summarizes the results. Even in this highly undersampled setting, LORE closely tracks the intrinsic rank across all tested values, while all baseline OE algorithms default to the embedding di- mension. Dim-CV not only is farther from the intrinsic rank than LORE but also takes considerably longer to run (note the log scaled y axis!) due to training multiple ordinal embeddings from hy- pothesis testing and cross validation. Also, LORE significantly outperforms baselines in test triplet accuracy, demonstrating robust ordinal embedding recovery even with noise, small amounts of data and realistic semantic structure in the percept while still reliably recovering the intrinsic rank.
5.5 LORE LEARNS LOW RANK AND ACCURATE REPRESENTATIONS ON CROWDSOURCED DATA
To test LORE in practical, noisy settings with unknown intrinsic rank, we evaluate it alongside base- lines on three representative crowdsourced human similarity datasets covering food images (Wilber et al. 2014), material images (Lagunas et al. 2011)) and car images (Kleindessner & Von Luxburg, 2017 These datasets differ in size, query semantics, and noise, reflecting the variety and challenges of real world ordinal data with results in Table 2 Further details are in Appendix E with additional real world dataset evaluations in Appendix K
All methods are trained on random sample comprising of 90% of the total triplets, Gaussian noise and the same embedding dimension. Across all datasets, LORE has comparable test triplet accuracy to existing methods, but uniquely yields a substantially lower rank (for e.g., Food- 100: LORE gets rank 3.3 vs. 15 for the others), suggesting a low rank structure. For materials, LORE gets both the highest triplet accuracy and a low rank structure of 2-3 dimensions verified in UMAP visualizations from (Lagunas et al. 2019). For cars, extreme noise is reflected in low accuracy for all methods yet LORE's s embeddings are more compact and do not degenerate to random chance like Dim-CV. There is considerable variance in time taken across methods and datasets due to the optimization characteristics of each method but LORE is the second fastest method after FORTE consistently.
Dim-CV performs poorly, likely due to its conservative approach in dimensionality selection by hypothesis testing, underfitting compared the other methods. Dim-CV is restricted
realistic dimensionality range (1-10) for data with unknown intrinsic rank. LORE not only learns low-rank representations but also maintains competitive triplet accuracy compared to methods that overparam- eterize the space, without needing dimensionality assumptions. These results show that only LORE recovers low rank structure (avoiding overparameterizing) without sacrificing significant accuracy (avoiding underfittin)) from real data, enabling practical perceptual modeling.
5.6 LORE'S LEARNED AXES ARE SEMANTICALLY INTERPRETABLE
Figure 5 shows that, without semantic supervision, LORE's s first three axes for Food- 100, the same embedding used for Table 2 each align with interpretable food properties: from sweet to savory (Axis 1), dense to light (Axis 2), and carb-rich to protein/vegetable (Axis 3). The last axis is slightly less coherent, as is expected for axes linked to smaller singular values. These results demonstrate that LORE actually recovers semantically meaningful latent dimensions while recovering a low rank embedding. Consequently, the axes are interpretable and this property is invaluable for scientific discovery where the subjective percept is not well understood. Additional interpretability results for the other methods are included in Appendix Fsswow that LORE's s axes are consistently more aligned with meaningful semantic concepts than every method except CKL which is comparable it.
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Table Comparison of OEs on Real Life Ordinal Datasets
Method
Food-100
Materials
Cars
Test Acc.
Time (8)
Test Acc.
Time (s)
Test Acc.
Time
Rank
Rank
Rank
(s)
LORE
82.45
3.3
6.64
84.08
2.23
5.77
52.12
3
4.45
(Ours)
+ 0.27
+0.47
+ 3.90
0.19
#0.43
3.37
+ 1.22
+0.45
+ 1.62
SOE
82.34
15
27.09
81.86
15
35.48
53.17
15.0
5.53
+ 0.32
+ 0.00
+ 1.38
+ 0.55
+ 0.0
1.30
+ 1.42
+ 0.0
+ 1.22
FORTE
81.73
15
6.34
79.35
15
3.74
52.91
15.0
0.85
+ 0.46
0.00
+ 0.52
0.77
+ 0.0
0.58
+ 0.84
0.0
+ 0.18
t-STE
82.79
15
40.93
83.44
15
27.15
53.70
15.0
15.13
+ 0.24
+ 0.00
+ 20.14
+ 0.49
+ 0.0
3.16
+ 1.15
0.0
+ 4.29
CKL
82.75
15
18.41
83.94
15
14.77
54.06
15.0
4.85
+ 0.20
0.00
+ 7.89
0.11
+ 0.0
1.79
+ 1.19
1 0.0
+ 0.39
Dim-CV
77.67
1.47
1721.9
78.10
1.0
1428.6
50.43
1.0
270.56
+ 0.02
+ 0.51
+ 26.71
3.79
+ 0.0
+ 32.84
+ 1.07
+ 0.0
+ 8.86
Cinnamon with Cream Cheese Glace
Barley with Apricots
Celery, Blue
Grasdhopper Pie
Yorkshine Pudding
Chickoea Sneck
Flatbread Pizzes
Hazzinu:
White Wine Souce
Salad Citrus Saled
Sweet
Piito Bean Salad with Tomatons, Red Onion,
2
Fresh and Mayer Lemon
Cot, and Nut Veogle Forger
Narinated Pork
Kattle Cooked
with Pine Nuts and
Carrets Parship
Auturnes Fanczan
Creme Bruice
Learned Axis Denser to Lighter
-3
-1
Firto Bean Salad with Tomatoss, Red Onion,
Fighting Allack with LowenCarth
immp and Bocon
Grocchi Mac and Blue
Regperjack
Sweet and
Shrimp
Learned Axis Caras to Nrgettn/Negttbblle
-2
Figure 5: LORE'S learned axes are semantically interpretable: Food groups as axis value varies for the first three learned axes of the LORE embedding learned on the Food- dataset. (Same embedding learned for Table 2)
DISCUSSION
In this work, we introduced LORE, a framework for jointly learning the intrinsic rank and the true perceptual latent space via an ordinal embedding. Our results show that LORE consistently recov- ers low-dimensional representations, with ranks that closely match ground truth while maintaining competitive test triplet accuracy. Moreover, we show that LORE does not underfit the perceptual space like Dim-CV does yet does not overparameterize the space like all other baselines do. On real crowdsourced data, LORE also uncovers interpretable axes aligned with meaningful semantic concepts, making subjective perceptual spaces easier analyze.
One limitation the absence of theoretical guarantees for exact rank recovery or optimal embed- dings. Our method empirically performs well, but its theoretical underpinnings remain open question as LORE's optimization is only guaranteed to reach stationary points, not global minima.
Future directions include developing theoretical guarantees and active learning methods collect perceptual data more efficiently. We hope our work inspires further applied and theoretical advances applying LORE to uncover the structure of perceptual spaces across range of domains.
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REPRODUCIBILITY STATEMENT
We have taken several steps to ensure the reproducibility of our work. All code and detailed docu- mentation for our artificial human experiments and crowdsourced human experiments included in the supplemental material, with full experimental configurations provided Appendix Mand Ap-
pendix
For the synthetic experiments and baseline comparisons, which require large scale parallelization and substantial computational resources, do not provide raw code. Instead, we describe in detail the procedures and parameter settings necessary to reproduce them in Appendix H
Our main theoretical result, establishing convergence of LORE local optimum, includes com- plete proof with all required assumptions in Appendix A To support practical use, we provide
demo (in the supplemental material) showing how LORE can be applied to new datasets, along with additional implementation details in Appendix B
The code to reproduce our results and a demo on how to use LORE are in iclr LORE's integration into cblearn (Künstle & von Luxburg, 2024), a Python package for ordinal embeddings and comparison- machine learning forthcoming. We believe these efforts will make our work fully reproducible and accessible to the community.
ACKNOWLEDCMENTS
We would like to sincerely thank Belén Martín FueefanUrrcela, Yenho Chen and Chiraag Kaushik for help- ful discussions and feedback on the paper.
This work was supported by the National Science Foundation under Grant No. CCF-2107455. Additionally, supported in part by the NIH BRAIN Initiative through NIMH grant R61MH1 33966, the National Center for Advancing Translational Sciences of the National Institutes of Health under Award Number ULITRO02378 and KL2TRO02381 I, and the Julian T. Hightower Chair at Georgia Tech. Vivek Anand was supported by the National Institutes of Health under the Data Science in Computational Neural Engineering Training Grant T32EB025816 1 and the National Defense Science and Engineering Graduate (NDSEG) Fellowship Program. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
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A PROOF FOR THEOREM 1
Theorem (LORE converges to a stationary point), The sequence of OEs generated by the LORE algorithm {Z+ 1k=,,,,,,.. converges. i.e.
|z
Zt F +00
(5)
Proof. We use the general framework for nonconvex Schatten Quasi-Norm optimization as seen in Sun et al., 2017) but check the specific conditions for the LORE objective.
Let us split up the objective as follows.
min
(Z)
Σ
log(1 + exp(1 d(Zo,: Zt.)) d(Zu.: Zz:3)) ZZI
(6)
(a.iJ)TT
min (N.d))
Σ
log(1 + exp(1 + d(Z,. Zi,)) d(Za.; Zz,1))
Σ
)g)o,(Z))
(a.JJ)TT
((2)
(7)
There are four assumptions we need to satisfy to apply the general result from (Sun et al. 2017).
A1. f is differentiable and has a Lipschitz gradient:
This stems from smoothing the triplet loss with the softplus function. The composition makes f differentiable everywhere except degenerate collapse points (which do arise with practical initializations). The log-sum-exp structure ensures (locally) Lipschitz gradi- ents (Chen et 2020).
A2. g concave, nondecreasing, and Lipschitz:
g(x) = x for x >> 0 has these properties.
A3. V is coercive:
For our objective. suppose |Z][ - oc. Then the sum of squared singular values diverges, at least one (Z)
As (Z) is the sum of all or (Z)P, P, and p >> 0, therefore have g( > 0o and thus (Z) + co.
A4. Vi has the Kurdyka Lojasciewicz (KL) property:
As established in (Bolte et al. 2010), sums of o-minimal (definable) functions, such loss and regularizer, possess the KL property.
With all required assumptions satisfied, Theorem of (Sun 2017 applies and guarantees:
Z ++ Z+Il co.
(8)
Therefore, the LORE algorithm converges to stationary point.
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B OPERATIONAL DETAILS FOR LORE
The optimization algorithm used for LORE is an adaptation of the original algorithm from (Sun al. 2017).
The function takes the initialized embedding Z" the regularization parameter A, the Lipschitz con- stant of V.()), μs, and the tolerance for convergence tol. The exact Lipschitz constant of s(-) is not known but was be empirically estimated to be strictly greater than 0.013 by the Power Iteration method. Therefore, we set μ 0.1 throughout our experiments. The algorithm initializes the ordinal embedding and iteratively updates it by minimizing the smoothed ordinal loss plus Schatten-p regu- larization. Each step performs proximal gradient update and singular value thresholding, repeating until convergence. Based on prior literature in the Schatten-p quasi-norm optimization literature (Wang et al. 2024 Sun et al. 2017; Lu et al. 2014), we fix p = 0.5 as has been shown to have good empirical results across various applications. Both and p hyperparameters that are needed to tuned for practitioner. Moreover, though À is a clear hyperparameter, our empirical results show that setting of À 0.01 works well across all of our experiments and would suggest practitioner start with this value. The tolerance tol is set to 10-5 throughout our experiments. More- over, cap the number of iterations to 1000 for all experiments ensure reasonable runtimes. If longer runtimes are acceptable, this cap can be increased to improve performance. However, this is usually very marginal.
If we consider the most standard operational setting, i.e. d' < N < T, then the time complexity of each iteration is O(d' (T Nd))). The dominating terms here the number of percepts and the number of triplets used with the most intensive operation is in line 8 where the gradient of subsequently performed. As result, LORE is scalable for higher N and d'. One does need to be careful as 7 could scale with O(N3) if many triplets are chosen which could slow down each iteration.
calculated and a singular value decomposition
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C
ADDITIONAL PLOTS FOR REGULARIZATION OF LORE
100
Test Triplet Accuracy
Number of Percepts
10
75
Rank
0.75
Intrinsic Rank
5
50
25
10
10"2
10
10
10
10
10
10
Figure 6: LORE has high test triplet accuracy and intrinsic rank recovery across various num- ber of percepts. (Left) Mean test triplet accuracy vs À for LORE as number of percepts varies. (Right) Mean measured rank vs À for LORE as number of percepts varies.
Figure 6 shows the test triplet accuracy and intrinsic rank recovery of LORE as the number of percepts varies. We see that with greater number of percepts rank recovery stays roughly constant whereas the test triplet accuracy increases significantly from 25-50 percepts. Baseline parameters are intrinsic rank 5, fraction of queries 0.1, noise 0.1.
15
Test Triplet Accuracy
1.0
0.5
Rank
0.75
0.1
Intrinsic Rank
0.05
0.01
0.5
10
10
10
100
10
10
10
0.0
Figure 7: LORE has high test triplet accuracy and intrinsic rank recovery across various noise levels. (Left) Mean test triplet accuracy vs À for LORE noise varies. (Right) Mean measured rank vs À for LORE as noise varies.
Figure 7 shows the test triplet accuracy and intrinsic rank recovery of LORE as the noise varies. We that with greater noise rank recovery and test triplet accuracy both decrease. There is dramatic drop in test triplet accuracy from to 5 noise. Baseline parameters are intrinsic rank = 5, number of percepts 50, fraction of queries 0.1.
15c
Test Triplet Accuracy
9
8
Intrinsic Rank
0.75
Rank
6
5
10
1
10
100
10-3
10-2
10-1
10
Figure 8: LORE has high test triplet accuracy and intrinsic rank recovery across various in- trinsic ranks. (Left) Mean test triplet accuracy vs À for LORE as intrinsic rank varies. (Right) Mean measured rank vs X for LORE intrinsic rank varies.
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Figure shows the test triplet accuracy and intrinsic rank recovery of LORE as the intrinsic rank varies. We see that for the same regularization level till approximately 8 dimensions. LORE can recover the rank. Baseline parameters number of percepts = 50, fraction of queries = 0.1, noise =.1..
These results, together with Figure 2 show that LORE is quite robust to the various knobs (noise, number of percepts, intrinsic rank and number of queries) that can be tuned for OE applications and that a regularization setting of À 0.01 learns an embedding with both high triplet accuracy yet
recover the intrinsic rank.
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D ADDITIONAL PLOTS COMPARING LORE TO BASELINES
1.00
Test Accuracy
10
LORE (ours)
0.75
Rank
SOE
Intrinsic
FORTE
5
0.50
25
50
75
100
25
50
75
100
Number of Percepts
Number of Percepts
Figure 9: Only LORE can recover the intrinsic rank while maintaining comparable test triplet accuracy. (Left) Mean test triplet accuracy vs number of percepts used for LORE and the baselines. (Right) Mean measured rank vs number of percepts used for LORE and the baselines.
Figureyshoove shows the triplet accuracy and intrinsic rank recovery of LORE and the baselines as the number of percepts varies. We see that with greater number of percepts rank recovery stays roughly constant, though spread decreases, for LORE from 25-50 percepts. Baselines again cannot recover the intrinsic rank at all. Test triplet accuracy increases from 25-50 percepts for all OE algorithms. Baseline parameters are intrinsic rank 5, fraction of queries = 0.I, noise 0.1.
1.00
leett Accuracy
10-
LORE (ours)
SOE
0.75
Intrinsic Rank
FORTE
0.50
10
10
10-2
10-1
Noise
Noise
Figure 10: Only LORE can recover the intrinsic rank while maintaining comparable test triplet accuracy. (Left) Mean test triplet accuracy vs noise used for LORE and the baselines. (Right) Mean measured rank noise used for LORE and the baselines.
Figure I0 shows the test triplet accuracy and intrinsic rank recovery of LORE and the baselines as the noise varies. We see that with greater noise, LORE is still able to recover the intrinsic rank though spread increases with noise from 1-5. The baselines cannot recover the intrinsic rank at all. Test triplet accuracy decreases with noise for all OE algorithms though LORE still performs the best. Baseline parameters are intrinsic rank 5, number of percepts 50, fraction of queries = 0.I.
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E
CROWDSOURCED DATASET DETAILS
Table 3: Details of Crowdsourced Datasets Used
Num. Percepts
Num. Triplets
Datasets
Triplet Type
Notes
Images
of foods.
Converted
similar-
ity triplets
Food-100
100
190,376
Compared to A, which is more similar, B or C?
using
(Wilber et al. 2014
'cblearn"
Künstle
von
Luxburg
2024).
Images of materials.
Converted to similar-
ity triplets
Materials
100
22,801
Compared to A, which is more similar, B C?
using
Lagunas et al. 2019
'cblearn'
Künstle
von Luxburg 2024).
Images of cars. Cen- tral triplets converted to similar- ity triplets using
Cars
7,097
Which A, B, C most central?
(Kleindessner & Von Luxburg 2017
'cblearn'
Künstle
&
von
Luxburg 2024).
Names of musicians. Converted to similar- ity triplets
Musicians Ellis et al. 2002
448
118,263
Compared to A. which is more similar, or C?
using
'cblearn'
Künstle
& von
Luxburg 2024).
Of these datasets, the Cars dataset is known be very noisy (Kleindessner & Von Luxburg. 2017 Vankadara et al. 2023). Food-100 has been used as a dataset to evaluate active querying methods (Canal et al. (020)). Musicians known be very undersampled in terms triplets compared the number of percepts and is not the desired operational setting of this work. A detailed characterization of the datasets seen Table
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F ADDITIONAL INTERPRETABILITY PLOTS
In this section we include interpretability plots for the other ordinal embedding methods on the Food-100 dataset. These plots analogous Figure in the main paper. The procedure that use to obtain these axes is to take the best performing ordinal embedding learned by each of these methods from Table 2 and then perform principal component analysis (PCA) the embedding to get the top three principal components. The reason for doing so is because ordinal embeddings cannot learn the directions of highest variance but only can learn an embedding that may be rotated, scaled or translated compared to the true perceptual space (Vankadara et al. 2023 Jain al. (016). Then, we vary the value of each principal component from its minimum to maximum value in the embedding and plot datapoints across each axis separately to see if there is any semantic meaning to the axis.
Cinnamann Buns Cheese Glaze
Roasted Autumn Saled
Apple Bacon Stutted Potarions
Mussels in Sauce
Grasshosper PM
Yarkshire Pudcing
Parn Perdu
Nathos
Ciria Saled
40
-30
-0
Yallow
-0
10
20
30
40
Fighting Snack Allask with Snacks
momaro Roasted Redd Pepcer, Vintt
Roasted
Autumn Salad
Black Sorbct
Rousced Snock
Thai Cocorut Noodlle Soup
Onion Tart
Codicall
Pain Pendu
Learned Axis 2: Lihter Denser
30
-20
-10
10
20
Roasted Brussels with Almords und
Sweeet Potato and Blue
Roasted Chickpua
That Coconut
Shrimp and bacon
Six imp
Chicken: Pupoerfack
Learndd Axis 3: Protein/Megerable Carbs
100
Figure 11: CKL's learned axes semantically interpretable: Food groups axis value varies for the first three learned axes of the CKL embedding learned on the Food- 100 dataset. These are very similar to LORE's seen Figure (Same embedding as one learned for Table 2
Morcccan Spring Vegetable
Sour Cream Chocolate
Grilled Asan Flank
Mazte Bacon
Creme Brutee
Pepperack
Candled
Learned Axts 1: Mostly Sweet Not Sweet
1.0
-0.5
0.0
0.5
1.0
Filhtng Attack with Lower-Cart
Brown and Veggiee Burger
Grilled fish with Citrus
Slew Cooker Bar becoa
Grilled Asien Flark Steak
Creamned Spinach
Unilled Corn
Zucchiri Soup
Apricot
Leamed Lighter Denser
-1.0
-0.5
0.0
0.5
1.0
Roasted Viggle Tistatas
Celery,
Gritted Asian Flank
Chocolate Valertine Cookdes
Ghorchi Mac and Blue Cheese
Pancetto
Gracshopper
Hazelnut
Maxican Com Salad
Ciaantr Gutche
Learned Axis
-1.0
-0.5
0.0
0.5
1.0
Figure 13: FORTE's learned axes are not fully semantically interpretable: Food groups as axis value varies for the first three learned axes of the FORTE embedding learned the Food-100 dataset. Only the first two dimensions are semantically interpretable but the third is not compared to LORE's Figure (Same embedding as one learned for Table 2).
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Ressbemyy
Ratebow truit
Purty Peanut Butter Cuckies with Chips
Busserry Gerpacho Mint
Snow Peas Pow Hirt
Skemers
Slow Cooker Beaf Stroganoff
Grilled Fota Salad
PasperjackMulle
Dip
Potatoss
Blondies
Bratwurst
Axis Mostly Sweet Not Sweet
0B40
00 oHID 0e @
ome
0.5
Lemon Ricetta Pound Cake
Homemaae:
Grasshopper Pie
Chedder
Flacbreed Pizoss
Apricnt Chicken
Griled Swied
White Sauce
Jeccs
Bisculs Pein Perdu
Learned Axis Mostly Sweet Not Sweet
000000
e< o0
ocmee
Whole Wheal
Spaghutt Italian Sausage, Ulies, and Basil Minalgrette
Autumn Parzandlaa Solad
Chickan with Mexican ConForty Cloves
Settle Cookee Popcom
Lacash Alttoeadd
Dut, but Slow Cuokar Buef Stroganoff
Veggie Burger
Learned Lighter Denser
0.6
0.4
0.2
0.0
0.2
0.4
Figure 12: SOE's learned axes are semantically interpretable but not parsimonious: Food groups axis value varies for the first three learned axes the SOE embedding learned on the Food-100 100 dataset. These contain two dimensions that LORE does but not all Figure 5 (Same em- bedding as one learned for Table
Coulffower Graiin with Sharp Chedsar and Parmessr
Roasted Panzonella Saled
Pancatta
Neppermint Brownies
White Sauce
Cliantro Quiche
The Uttimate Onion Tart
Raspberry Sortet
Chocolate Cookies
Feta Salao
oae eo
Not
Learned Axis 1:
Chunky
Sweet Porato fanozzhi Mac and Cheese
Cinnamen Apples
with Hine Mint
Itomeeade
Slow Cupker Stroghnott
Mayonnatze
Pumpkc
Amstrictana
Puddiig
600 00
BOO
anmeco
Learned Axis 2:
-2 Bean and with
Cannellnii
Moroccan Spriing Vegetaole Tagine
Chocolate Hazelnut
Shrimp and Bacon
Chicken Florentine
Lawash Flatoread
Tomotoes and
Sweet and
Leamnd Axis
-4
-2
Figure 14: t-STE's learned axes are not fully semantically interpretable: Food groups as axis value varies for the first three learned axes of the t-STE embedding learned the Food-100 dataset. Only the first dimension is semantically interpretable but the second and third are not compared to LORE's Figure (Same embedding
learned for Table 2).
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Vellow Salad
Skinny Red Velvet Cupcakes
Black Raspberry Sorbet
Gnocchi Peaper. and Blux Feta, and Cheese
Thai Cocors Grilled Asian Flank Steek
Mussels White Wina Sauce
Eggnog Ice Cream
Yorkshire Pudding
Chicken Noodle Sou
Mint
Learned Axis Sweet
-0.2
0.2
0.4
0.6
Pumpkin- Low Sow--ugar Celery, Blue ChocolstePumpkin aniCheese and Hazelnut Apple
Roasted AutumnShrimp and Bacan
Crame
Apricat
Chunk
Leamed Axis 2:
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
Figure 15: Dim-CV's learned axes are not fully semantically interpretable: Food groups as axis value varies for the first two learned axes of the Dim-CV embedding (chosen after retraining and dimensionality selection) learned on the Food- 100 dataset. Only the first dimension is semantically interpretable but the second is not compared to Figure 5 (Same embedding as one learned for Ta- ble D Note that Dim-CV only learns two dimensions so only show two here.
From the above plots, we see that only CKL is able to learn all the three interpretable dimensions that LORE is able to learn. Both FORTE and TSTE are able learn but not all three. While SOE does learn all three interpretable dimensions, it splits sweetness to not sweet
interpretable dimensions
dimensions rather than learns a low rank embedding (only two dimensions), the second axis is not interpretable at all. Moreover, as seen in Table 2 it takes an order of magnitude longer to train. Though both CKL and LORE are able to learn comparable interpretable dimensions, LORE is superior as it also learns lower rank representation as well indicating that it does capture only the most important dimensions of the perceptual space without underfitting or overparameterizing.
LORE and CKL do. Dim-CV meanwhile though
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G EXAMINING THE EFFECT OF p ON LORE
In the following results we examine the effect of the p in the ordinal embedding and rank recovery for LORE. Specifically, we vary p with three values: 0.1, 0.5 (what we use in the paper) and (equivalent to the nuclear norm). Like in we vary the number of percepts, noise and intrinsic rank while keeping the other parameters constant to see how affects the test triplet accuracy and rank recovery albeit at reduced scale due to the computational cost of experiments. We include code for this experiment despite the additional time needed to run it. We find that results
has quite similar 0.5 albeit with reduced optimal regularization level of around 10-2.5 However, we notice that the leeway to choose the regularization parameter is much higher in absolute terms the performance of p 1 drops off more sharply
for p 0.5 than p optimal regularization level. This is especially apparent for high noise levels. p= 0.1 however is essentially useless obtain lower rank solutions, at least for the range of regularization that we have examined. The baseline parameters we use for all of them, unless we are varying them, are number of percepts 50, intrinsic rank 5, fraction of queries = 0. and noise
deviate from the
1.0
15
1.0
Test Accuracy
10
0.75
Rank
5-
0.1
0.5
10
10
10
10"
10-4
-2
10
10a
Figure 16: Both p 0.5 and p 1.0 have stable rank recovery settings.. (Left) Mean test triplet accuracy vs À for LORE. (Right) Mean measured rank vs À for LORE.
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1.0
15p
Accuracy
p=0.1
Rank
0.75
5
10
10-2
10
10
10
10
109
1.0
0.2
Tratsion Tripets
Accuracy
10
0.1
p=0.5
Rank
0.08
0.01
102
1.0
Accuracy
10c
Rank
p=1.0
10
102
Figure 17: p 0.5 has the most stable rank recovery settings as fraction of train triplets varies. (Left) Mean test triplet accuracy X for LORE fraction of triplets used varies. (Right) Mean measured rank vs for LORE fraction of triplets used varies.
1.0
Bank
p=0.1
0.75
54
1
0.5
10
10
10c
10
10
10-1
15
5.0
10
Rank
0=0.5
0.75
0.05
0.01
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10
10-1
10P
10
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30
0.0
1.0
Accuracy
Bank
p=1.0
0.75
5
0.5
30
10
30
100
10:3
10
16.3
1.0 have stable rank recovery settings noise varies. (Left) Mean test triplet accuracy vs for LORE as noise varies. (Right) Mean measured rank vs for LORE as noise varies.
Figure 20: Both 0.5 and
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10
Tect Accunay
10
Rank
p=0.1
10
10
100
10
10
10
1.0g
15
Number of Percepts
Redt Accuracy
10-
0=0.5
0,75
Rank
50
10
10
10
10"
1.0
15
Texz Accurary
Rank
p=1.0
0.75
0.5
10
10
10
100
10
10
10"1
10
Figure 18: 0.5 has the most stable rank recovery settings number of percepts varies. (Left) Mean test triplet accuracy vs for LORE as number of percepts varies. (Right) Mean mea- sured rank vs À for LORE as number of percepts varies.
1.0
Test Accuracy
Rank
p=0.1
0.75
54
1
0.5
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20
Intrinice Bankc
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0=0.5
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10
109
10-3
10
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10
1.0
Test Accuracy
20
Rank
p=1.0
10
20"2
100
10
10"
Figure 19: Both p = 0.5 and = 1.0 have stable rank recovery settings as intrinsic rank varies. (Left) Mean test triplet accuracy À for LORE intrinsic rank varies. (Right) Mean measured rank vs for LORE as intrinsic rank varies.
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H EXPERIMENTAL SETUP FOR SECTION 5.2 AND SECTION 5.3
This experiment was performed on a SLURM server with over 30 GPUs of varying quality and compute power. We do not include the scripts used to run those experiments as they are highly complex due to parallelism and take too long to run (over days). However, quick rundown of the experiment is given below.
A grid search over all the following parameters was performed for these experiments. The grid search was performed in parallel over 30 GPUs. Each experiment was run for 30 runs with different random seeds. The results were averaged over the 30 runs and the standard deviation was calculated.
In our experiments, the various knobs we tune are as follows.
Number of Percepts (N): We vary it from [25, 50, 75, 100] and 50 Percepts as a default. We do not increase the number of percepts beyond 100 as the number of queries increases combinatorially, Additionally, this is not practical number of percepts to collect for perceptual experiments.
True Dimension (d): We vary it from [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] and use 5 as default. We do not examine over 10 dimensions as it is not possible to resolve that many dimensions without increasing the number of percepts due to the curse of dimen- sionality (Bishop & Nasrabadi 2006).
Fraction of Queries used: we vary it from [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9] and use 0.1 as default.
Noise (a²): we vary it from [0, 0.01, 0.05, 0.1, 0.5, 1.0, 5. 0] and use 0.1 as a default.
Regularization (A): This is only for LORE but we vary with [0, 0.001, 0. 00158489, 0.00251189, 0.00398137, 0.00630957, 007666665 0. 0.00936329, 0.01, 0.01140625, 0. 0.01389495, 0.01692667, 0.02661986, 0.02511886, 0.0305995, 0.03727594, 0.0454091, 0. 0.05531681, 0.06738627, 0.08208914, 0.1, 0.21544347,
0 0.46415888, 1.] and use 0.01 as default.
Embedding Dimension (d'): This is the dimension of the embedding we are trying to learn. This is only for the baselines other than LORE. We vary it from [1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15] and use as a default.
The metries
follows
Test Triplet Accuracy: The accuracy of the test triplets on the test set. This is the main metric we use to measure performance.
Measured Rank: The rank of the embedding matrix. This is measure of how well the algorithm is able to recover the intrinsic rank of the data. We measure this by taking the SVD of the embedding matrix and counting the number of non-zero singular values. Specifically, we use the rank function from the numpy library to compute the rank of the embedding matrix.
The PSNR is measure of the quality of the recovered matrix. However, note that the recovered embedding matrix has to be aligned to the true percepts matrix compute the PSNR. The specific formulation is described Appendix N
Peak Signal to Noise Ratio:
Normalized Procrustes Distance:
The NPD is measure of how well the recovered ma- trix matches the true matrix up to rotation, scaling and translation. To perform procrustes analysis, true percepts P E IR N xd and the computed embedding Z E Nx must be the same shape. Therefore, two matrices have the same shape. The specific formulation is described in Appendix N
the same subspace alignment technique to ensure that the
should be noted that test triplet accuracy and measured rank are the main metrics we use to measure performance the other metrics require knowledge of the percepts P which is not known in practice.
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I EXPERIMENTAL SETUP FOR SECTION 5.4
50 random foods were chosen from the Food-100 dataset (Wilber et al. 2014). This was run on server with RTX3080 GPU and 128 GB of RAM. The names of the specific percepts are as follows.
[' Cinnamon Swirl Buns with Cream Cheese Glaze',
Shrimp and Bacon Risotto', Shrimp Cocktail',
Homemade Cracker Jacks',
Creme Brulee French Toast', Red Lobster Cheddar Bay Biscuits', Apple Bacon Stuffed Sweet Potatces', Sweet-ande -aieett-ad-Souu Sour Chicken', Pumokin-Chocolate Chunk Pancakes', Chocolate Hazelnut Biscotti', Eggnog Ice Cream', Celery, Blue Cheese and Hazelnut Salad', Shredded and Roasted Brussels Sprouts with Almonds and Parmesan', LLw Sugar Pumpkin and Apple Crumble', Roasted Sweet Potatoes Recipe with Double Truffle Flavor and Parmesan', Chicken Florentine Bowtle Pasta', White Whole Wheat Pizza Dough', Chervil Mayonnaise',
Pork Tenderloin in Tomatillo Sauce', Yellow Tomato Salad with Roasted Red Pepper, Feta, and Mint', Daisy Brand Sour Cream Chocolate Cake', Shredded Brussels Sprouts Apples', Mexican Corn Salad', Potato Skins',
Caramel Kettle Cooked Popcorn',
Roasted Garlic Veggie Tostadas', Pan Seared Scallops with Baby Greens and Citrus Mojo Vinaigrette', LLmon Cranberry Scones', Warm Butternut and Chickpea Salad with Tahini Dressing', "Fighting Off That Snack Attack with Lower-Carb Snacks', Chicken with Forty Cloves Garlic',
Edna Mae' s Sour Cream Pancakes', Sweet Potato Gnocchi Mac and Blue Cheese', Yorkshire Pudding', Luscious Lemon Squares', "Japanese Pizza', 'Grilled Asparagus & Feta Salad',
Grilled Corn Salad',
Garllic Meatball Pasta', Roasted Autumn Panzanella Salad',
Coconut Marinated Pork Tenderloin', Black Raspberry Sorbet', 'Mini Whole Wheat BBQ Chicken Calzones', Mussels in White Wine Sauce', Brown Rice, Oat, and Nut Veggie Burger', Dark Chocolate Cookies', Citruu Salad',
RRastedd Carrots & Parsnip Puree', South African Cheese, Grilled Onion & Tomato Panini (Braaibroodjie)) 'Pinto Bean Salad with Avocado, Tomatoes, Red Onion, and Cilantro')
These names are passed to the SBERT library (Reimers & Gurevych, 2019) with the "all-mpnet- base-v2" model to get 768 dimensional LLM embedding. To simulate various possible intrinsic
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ranks, use the truncated singular value decomposition constrain the "true" perceptual repre- sentations foods to intrinsic ranks 1-10. Specifically, the truncated SVD is the following.
The singular value decomposition of matrix P' E RNxx is given by
P' UEV
Here U E RNNN,, S E RN and V E R 768x 768 U and V have orthonormal columns and E is a diagonal matrix with singular values 01 2 02 2 ... 2 ON > 0. The intrinsic rank of the matrix the number of non-zero singular values, which in this case is N before truncation. We can truncate the SVD to fix an intrinsic rank of d.
If U Nxx and Va I768xd are the first d columns of U and V respectively, Eg Rdxd with the biggest d singular values in the diagonal entries and otherwise then we can write the truncated SVD of P' to get the our "true" perceptual representation of the foods
P' V.E.V
(10)
Then, we query just 5% of the total possible triplets (2940 out of possible 58800) with 0.1 vari- ance Gaussian noise added the triplet comparisons to simulate the noise that humans have when answering triplet queries. 5% of the total queries is reasonable setting common to most perceptual scaling experiments as it is usually the bare minimum of queries needed to fit a good embedding (Könstle et al.) 2022); Vankadara et al. 2023). We repeat this sampling and query answer simula- tion process thirty times, independently with various random seeds, and train all of the the various OE algorithms. This is to obtain pathological training sets. The metrics we measure are as follows.
robust results and avoid the effects of bad initializations or
Test Triplet Accuracy: The accuracy of the test triplets on the test set (fixed at 3000 queries not in the train set and chosen at random). This is the main metric we use to measure performance.
Measured Rank: The rank of the embedding matrix. This is a measure of how well the algorithm able to recover the intrinsic rank of the data. We measure this by taking the SVD of the embedding matrix and counting the number of non-zero singular values. Specifically,, we use the rank function from the numpy library to compute the rank of the embedding matrix.
Peak Signal to Noise Ratio: The PSNR is measure of the quality of the recovered matrix. However, note that the recovered embedding matrix has to be aligned the true percepts matrix to compute the PSNR. The specific formulation is described in Appendix N (We do not report these in the paper)
Normalized Procrustes Distance: The NPD is measure of how well the recovered ma- trix matches the true matrix up to rotation, scaling and translation. To perform procrustes analysis, true percepts P E N and the computed embedding Z E Nxd must be the same shape. Therefore, we use the same subspace alignment technique to ensure that the two matrices have the same shape. The specific formulation is described in Appendix N (We do not report these in the paper)
Code for this experiment included the supplemental material.
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J EXPERIMENTAL SETUP FOR SECTION 5.5
This was run on a server with 1 RTX3080 GPU and 128 GB of RAM.
For LORE, set the regularization parameter, X, to 0.01. For all OE methods, set the number of dimensions of the OE, d', to 15.
Note that for this experiment unlike in Appendix [ we do not have access the true percepts P and therefore cannot compute the PSNR or NPD. We only report the test triplet accuracy and measured rank. For Dim-CV we use total of cross validation folds but do not use multiple initializations due to computational constraints. As it is, Dim-CV is already two orders of magnitude slower than all of the OE algorithms due the additional burden of training multiple embeddings due to the cross validation procedure. From our observations, increasing the number of cross validation folds and using multiple initializations reduces the standard deviation the Dim-CV test triplet accuracy and rank but not the mean. Time however, increases linearly with the number of cross validation folds and initializations.
Code for this experiment is included in the supplemental material.
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K ADDITIONAL CROWDSOURCED DATASET EXPERIMENTS
The results on one more crowdsourced real life dataset, the musicians dataset (Ells et al. 2002) seen in Table This dataset contains 448 percepts (musicians) and 118.263 triplet comparisons collected from human annotators. This is considerably undersampled in terms of triplets compared the other datasets included in the main text. For example, Food-100 contains more triplets even though it has fewer percepts. The results are shown in the figure below. We do not run the Dim- CV due computational constraints from the hypothesis testing procedure. We anticipate would take "7000 seconds per iteration to run Dim-CV on this dataset given that it took "1700 seconds per iteration on the Food-100 dataset which has roughly one fourth the number of percepts. Therefore, exclude from this experiment. We set d' 30 as the embedding dimension for all OE methods as this has considerably more number of percepts than the other datasets.
Table 4: Comparison of OEs on Musicians Dataset
Method
Musicians
Metric Std
Test Acc.
Rank
Time (s)
LORE (Ours)
75.63 + 0.94
27.8 + 0.55
13.82 + 9.72
SOE
81.41 = 0.93
30 0.0
28.45 = 2.20
FORTE
69.94 = 1.61
30 = 0.0
8.63 = 2.79
t-STE
79.49 1.52
30 0.0
98.97 81.26
CKL
78.05 + 0.96
30 -0=00 0.0
24.3 + 10.51
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L SCALABILITY OF LORE
1000
Iterations
100
230
500
500
1000
2500
2000
Time (seconds)
25
500
Number of Percepts
Number of Triplets
Intrinsic Rank
A
B
C
Figure 21: LORE remains scalable as dataset parameters change: (A) Time and Number of Iterations as Number of Percepts Varies (log scaled axis) (B) Time and Number of Iterations as Number of Triplets Varies (log scaled x axis) (C) Time and Number of Iterations as Intrinsic Rank Varies. Error bars indicate + two standard deviations over random seeds of the generative process for different datasets. Baseline parameters are intrinsic rank 5, number of percepts 50, number of triplets 2500.
For our empirical implementation in this paper, we cap the number of iterations of LORE to 1000. In this experiment, we fix the noise to a moderate noise of 0.1 variance sampled from a Gaussian distribution. As seen in Figure 21 which varies dataset characteristic while keeping the others constant, the number of iterations correlates almost perfectly with the time taken to run LORE. Therefore, if LORE converges in fewer iterations, it will take less time to run and the length of each iteration is roughly constant for given dataset. We see that as number of percepts increases, the number of iterations increases until it hits the 1000 iteration cap. This likely due to the fact that as the number of percepts increases, the optimization problem becomes more ambiguous as Z increases in size. As the number of triplets increases, the number iterations decreases likely because there are more constraints to guide the optimization. Finally, the intrinsic rank does not seem to have significant effect on the number of iterations needed converge.
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M CONVERGENCE OF LORE
20
Number of Percepts
0.3
Number of Triplets
3--
+5 tanteen
Intrinsic Rank
1:4
6.4
Low
Baseline
High
Figure 22: LORE convergences smoothly despite differrnt initializations and nonconvexity: Objective function values as LORE trains for 30 random seeds for the same true perceptual space. Each color is a separate seed run. Number of Percepts, Number of Triplets and Intrinsic Rank are varied individually. Baseline parameters percepts = 50, number triplets 2500.
space defined by intrinsic rank 5, number of
To visually examine the convergence of LORE we run an experiment where we fix the true percep- tual space and vary the random seed the initialization of LORE. We see in Figure that despite the non-convexity of the optimization problem, LORE converges smoothly across different random initializations. We vary one dataset characteristic at time while keeping the others constant like in the previous section. We see that the number of percepts increases (25, 50, 500), the objective function convergence takes longer likely due to the increased ambiguity of the optimization problem the size of Z increases. As the number of triplets increases (250, 2500, 5000), convergence is faster likely due to the increased constraints on the optimization problem. Finally, the intrinsic rank (2, 5, 10) does not seem have a significant effect on convergence speed.
Taken together with Appendix these results indicate that LORE smoothly converges despite the non-convexity of the optimization problem and that the number of iterations taken converge scales reasonably with dataset parameters. This empirically shows the provable convergence to stationary point that we show Appendix A This result in combination with the fact that local minima for ordinal embedding problems are often good solutions (Vankadara et al., 2023) indicates that LORE a scalable and practical algorithm for ordinal embedding in real world settings and that the theory supports the empirical findings.
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N
FORMULATION OF OTHER METRICS
Code for all of these implementations is included in the supplemental material.
N.1 SUBSPACE ALIGNMENT
To perform procrustes analysis, true percepts R Nxd and the computed embedding RNxx must be the same shape.
Specifically we compute
P and Ze = 1Nf2
(11)
Then, compute the tikhonov regularized projection matrix to prevent numerical instability due ill conditioning, We use regularization parameter of 17 = Le 3 3.
A (Z. + 7 'z P.
(12)
Then, we can get the aligned ordinal embedding Zaligned Z.A + IN/p-
N.2 NORMALIZED PROCRUSTES DISTANCE
Now that we have an aligned matrix the same shape as P, the normalized procrustes distance be- tween the aligned embedding and the true percepts can be computed
P
Zuligned IF
Normalized Procrustes Distance
(13)
Ze
N.3 PEAK SIGNAL TO NOISE RATIO
The Peak Signal to Noise Ratio (PSNR) is measure of the quality of the recovered matrix and defined as 20 20 (aaiiiieddd P)( where P is the true matrix.
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O LLM USAGE
In this work, we leverage the use of large language models for two purposes. (1) to refine the writing by eliminating grammatical errors and improving flow. However, these were only used the individual paragraph level rather than whole sections and (2) to discover similar papers during the literature review for the related work. Specifically, we searched for terms like "distance metric learning", "contrastive learning", "psychophysical scaling" etc.
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Reviews:
[{"review_id": "qQ8nA5pHod", "reviewer": "Reviewer_sNLD", "summary": "The paper claims that all existing OE approaches are based on pre-specified embedding dimensions, which may lead to some problems. Meanwhile, the paper emphasizes the advantages of low intrinsic dimensional embedding--easier to interpret, less computationally intensive, while existing OE approaches based on pre-specified embedding dimensions often result in high-dimensional embeddings. \nBased on this, the paper introduces LORE (Low Rank Ordinal Embedding), a novel method for ordinal embedding that jointly learns both the low-dimensional embedding and **its intrinsic dimensionality** from noisy triplet comparisons.\nFurthermore, the paper establishes an efficient optimization strategy based on iteratively reweighted minimization and provides a scalable algorithm suitable for large-scale perceptual similarity data.\nAnd the paper validates the effectiveness and efficiency of LOPE through an extensive evaluation", "questions": "1. If the embeddings of other OE methods are not interpretable, the differences between other methods and LORE should be compared in Figure 5.\n2. The factors that affect computational efficiency (time metric) may need to be explained in order for readers to understand why the time differences of methods such as SOE/t-STE can be so significant at the same rank.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The explanation of the background and significance of the problem is very clear, and the problem to be solved is very meaningful.\n2. LORE is effective in reliably overlooking the intrinsic dimensionality and demonstrates the interpretability of low dimensional representations in semantics, which is helpful for solving problems in psychology, neuroscience, and social science.\n3. The theoretical explanation is very rigorous.", "weaknesses": "1. More new methods should be compared, and more datasets should be compared, especially considering that SOE and t-STE are both methods from 2014. This may lead to doubts about the performance of LORE.\n2. On the accuracy metric, which may be the most important metric, LOPE is not always optimal or even suboptimal.\n3. A low rank does not necessarily mean an improvement in method performance, so more explanation is need.\n4. For the metric of computational efficiency(time), low dimensional embedding is not the only solution(eg: FORTE vs. LORE), so the advantages of LORE should be further explained.\n5. Following 3, if the embeddings of other OE methods are not interpretable, the differences between other methods and LORE should be compared in Figure 5.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SIY4AD16D9", "reviewer": "Reviewer_86F2", "summary": "The paper proposes LORE (Low Rank Ordinal Embedding), an ordinal-embedding framework that jointly learns (i) an embedding that satisfies triplet comparisons and (ii) the intrinsic dimensionality (rank) of the latent perceptual space. The key idea is to regularize the embedding matrix with a nonconvex Schatten-p quasi-norm (with smoothing of the triplet loss), optimized via an iteratively reweighted scheme that is shown to converge to a stationary point. Experiments indicate that LORE achieves comparable triplet accuracy while discovering substantially lower-rank solutions.", "questions": "1 Can you provide calibration evidence to show that the method neither hides structure nor inflates rank?\n\n2 Do you have experiments where the true rank lies outside the candidate set used by Künstle et al. (2022)? How often does this occur, and what is the performance degradation?\n\n3 For equal target triplet accuracy, what is the cost for LORE vs. training multiple embeddings as in Künstle et al. (2022)? \n\n4 Why is there no quantitative comparison to Künstle et al. (2022)? \n\n5 How does LORE differ conceptually and empirically from recent works?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1 Tackles a long-standing limitation of ordinal embedding, i.e., choosing the dimensionality, by jointly inferring rank and coordinates, rather than grid-searching over dimensions. \n\n2 Uses Schatten-p regularization (p∈(0,1)) to promote low rank, with a softplus-smoothed triplet loss and an iteratively reweighted algorithm; provides a convergence-to-stationary-point guarantee and implementation details. \n\n3 Experiments indicate that LORE achieves comparable triplet accuracy while discovering substantially lower-rank solutions.", "weaknesses": "1 The theory ensures convergence to a stationary point, but not global minima or exact rank identification; this is acknowledged as a limitation.\n\n2 The paper argues that LORE uncovers the intrinsic dimensionality without under- or over-estimating it. As stated, this reads as a subjective claim. Please provide stronger evidence to demonstrate that the method does not “mask” latent structure or inflate rank.\n\n3 This paper claims that Künstle et al. (2022) require specifying plausible dimensionalities, risking misspecification and loss of power if the true rank lies outside those bounds. Do you have experiments showing this failure mode and quantifying how often it occurs under realistic sampling/noise? \n\n4 The paper states that training separate embeddings per hypothesized rank (as in Künstle et al., 2022) is computationally prohibitive. Please report the total cost to reach the same triplet accuracy for both methods.\n\n5 Missing direct comparison to Künstle et al. (2022).\n\n6 Literature coverage is dated. The citations lean heavily on pre-2022 works and omit several recent, directly relevant works.\n\n[1] Künstle D E. Machine Learning for Psychophysical Scaling with Ordinal Comparisons[D]. Eberhard Karls Universität Tübingen, 2024.\n[2] Huber L S, Künstle D E, Reuter K. Tracing truth through conceptual scaling: Mapping people’s understanding of abstract concepts[J]. 2024.\n[3] Sauer Y, Künstle D E, Wichmann F A, et al. An objective measurement approach to quantify the perceived distortions of spectacle lenses[J]. Scientific Reports, 2024, 14(1): 3967.\n[4] Huber L S, Künstle D E, Reuter K. Tracing truth through conceptual scaling[J]. Cognition, 2026, 266: 106321.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qboVUjx1Z0", "reviewer": "Reviewer_inQi", "summary": "The paper introduces LORE (LOw Rank Embedding), a novel and scalable framework designed to jointly learn the intrinsic dimensionality ($d$) and the optimal relative structure of perceptual spaces from noisy ordinal data (triplet comparisons of the form \"A is more similar to B than C\"). Addressing a fundamental limitation of existing Ordinal Embedding (OE) methods that rely on pre-defined or estimated dimensions, LORE leverages a low-rank constraint on the embedding matrix $Z$ and employs a highly non-convex Schatten quasi-norm as a regularizer to promote the discovery of the true intrinsic dimensionality. The optimization is handled by an effective iteratively reweighted algorithm with provable convergence guarantees. Extensive experiments on synthetic data, simulated perceptual spaces, and real-world crowdsourced datasets demonstrate that LORE successfully recovers the true intrinsic rank, achieves competitive triplet accuracy, and yields semantically interpretable embedding axes.", "questions": "1. The optimization relies on the non-convex Schatten quasi-norm. Could the authors provide a more detailed analysis or empirical evidence (e.g., through multiple restarts with different random initializations) showing the consistency and quality of the stationary points reached by the algorithm?\n\n2. The paper uses a simulated perceptual space derived from a large language model (LLM) embedding. Can the authors provide more intuition or validation for why the LLM's embedding space represents a \"true perceptual $d$-dimensional space\" that LORE is attempting to recover, and how the inherent noise was modeled in this specific experiment?\n\n3. Regarding the runtime complexity, how does the convergence speed (number of iterations for Algorithm 1) of LORE change as the total number of triplets ($T$) and the true intrinsic dimension ($d$) scale?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper addresses the critical and underexplored problem of jointly discovering the intrinsic dimensionality and relative structure in perceptual spaces, which is a key limitation of prior Ordinal Embedding (OE) methods. The introduction of the low-rank constraint via the non-convex Schatten quasi-norm is highly novel within the OE literature.\n\n2. Unlike many empirical approaches, LORE provides a convergence theorem (Theorem 1, page 5) for its optimization objective and the proposed iterative reweighted algorithm. This rigorous theoretical foundation significantly strengthens the paper's contribution.\n\n3. The experiments are thorough and persuasive. LORE successfully recovers the true intrinsic rank in synthetic and simulated LLM perceptual spaces (Figure 4) where other baselines fail. On real crowdsourced data (Food-100, Musicians, Cars), it maintains high triplet accuracy while achieving significantly lower rank embeddings compared to SOTA OE methods (Table 5).\n\n4. The learned embedding axes (Figure 5) are shown to be semantically interpretable (e.g., \"Sweet to Savory,\" \"Learned Axis 1\"), offering valuable insights into the underlying perceptual characteristics of the data, which is highly beneficial for discovery tasks.", "weaknesses": "1. While the paper provides a convergence theorem, the optimization objective $\\min \\Psi(Z)$ remains highly non-convex. The analysis primarily focuses on convergence to a stationary point, which may not always be the globally optimal solution. A more in-depth discussion on the practical robustness to initialization and the likelihood of escaping poor local minima would be beneficial.\n\n2. The LORE objective function includes several regularization parameters ($\\lambda, \\tau, \\mu$). Although Figure 2 demonstrates stability across a range of $\\lambda$ values for a fixed $\\tau$, a full exploration of the joint sensitivity of $\\lambda$ and $\\tau$ is absent. These parameters are crucial for balancing triplet accuracy and rank recovery, and their interplay needs more detailed investigation.\n\n3. The paper should explicitly discuss the cases where the intrinsic rank $d$ may not be an integer (e.g., fractional dimensionality in complex manifold structures) and whether LORE's reliance on a rank constraint limits its ability to fully capture these more intricate data structures.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xXm1PILwIx", "reviewer": "Reviewer_kWHX", "summary": "This paper presents LORE, a framework that jointly learns ordinal embeddings and their intrinsic dimensionality using a nonconvex Schatten-p regularization and an iteratively reweighted optimization algorithm. Experiments on synthetic and real perceptual datasets show that LORE can automatically recover low-rank, interpretable embeddings with competitive triplet accuracy.", "questions": "1. Under what data or noise conditions can the Schatten-p regularization reliably recover the intrinsic rank?\n\n2. How sensitive is LORE to the choices of \\lambda and p?\n\n3. Could the authors show more evidence of the optimization’s empirical behavior, such as convergence stability or runtime?\n\n4. All experiments are based on psychophysical or human-judgment data (food, music, cars). Have the authors tested, or plan to test, the method on non-perceptual ordinal datasets (e.g., image or text similarity)?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The paper presents an interesting framework (LORE) that aims to jointly learn ordinal embeddings and their intrinsic dimensionality, addressing a recognized limitation of existing approaches.\n\n2. The proposed optimization procedure is clearly described and includes a convergence argument, suggesting technical soundness.\n\n3. Experimental results across synthetic and real perceptual datasets provide encouraging evidence that LORE can recover compact and interpretable embeddings.", "weaknesses": "1. The paper lacks a theoretical analysis explaining under what conditions the Schatten-p regularization can correctly recover the intrinsic rank, which limits the strength of its main claim.\n\n2. The iteratively reweighted optimization is presented formally but lacks practical insight; for example, the paper does not show convergence curves, runtime comparisons, or how initialization influences final embeddings.\n\n3. The figures lack sufficient information for interpretation. Several plots (e.g., Figure 2 and Figure 3) omit axis labels or error ranges, and some results do not specify experimental settings or data sources, reducing the clarity and comparability of the findings.\n\n4. The study provides limited discussion of hyperparameter sensitivity, particularly the effects of λ and p on performance and rank estimation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)