[Paper Review] LORE: Jointly Learning the Intrinsic Dimensionality and Relative Similarity Structure From Ordinal Data

Vivek Anand, Alec Helbling+4 more|arXiv (Cornell University)|Feb 4, 2026

Aesthetic Perception and Analysis0 citations

TL;DR

LORE jointly learns both the embedding and the intrinsic dimensionality from noisy triplet comparisons by regularizing with the nonconvex Schatten-p quasi-norm, with convergence guarantees to stationary points.

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.

Motivation & Objective

• Motivate learning subjective percepts from ordinal data and avoid pre-specifying embedding dimensionality.
• Develop a scalable ordinal embedding method that jointly recovers the embedding and its intrinsic rank.
• Regularize to balance triplet accuracy with representation compactness to avoid underfitting and overparameterization.
• Provide convergence guarantees for the optimization procedure and demonstrate interpretability of learned axes.

Proposed method

• Formulate an ordinal embedding objective with triplet loss plus a Schatten-p quasi-norm regularizer on Z.
• Replace the nuclear norm with the nonconvex Schatten-p quasi-norm to encourage rank recovery.
• Smooth the hinge triplet loss with a softplus to make the objective differentiable and amenable to optimization.
• Solve the nonconvex objective via an iteratively reweighted algorithm that updates Z and the regularization weights at each step.
• Provide convergence guarantees: the sequence of embeddings converges to a stationary point.

Experimental results

Research questions

• RQ1 Can we jointly infer the embedding and its intrinsic dimensionality from ordinal triplet data?
• RQ2 Does Schatten-p regularization improve intrinsic rank recovery compared to traditional nuclear norm regularization?
• RQ3 Is the proposed iterative optimization robust and scalable to large datasets while preserving triplet accuracy?
• RQ4 Do learned low-rank embeddings yield semantically interpretable axes on real perceptual data?

Key findings

• LORE consistently recovers low-rank embeddings that match the true intrinsic dimensionality across synthetic and real data.
• LORE achieves competitive test triplet accuracy while recovering substantially lower rank than baselines.
• LORE outperforms baselines in intrinsic rank recovery across varying query fractions and noise levels.
• On crowdsourced real data, LORE learns semantically interpretable axes aligned with meaningful perceptual concepts.
• The method scales with complexity O(d′(T+Nd′)) per iteration and converges to stationary points.