LORE: Jointly Learning the Intrinsic Dimensionality and Relative Similarity Structure From Ordinal Data

2026-02-04

AI Summary

This work addresses the challenge of recovering the intrinsic dimensionality and similarity structure of subjective perceptual spaces—such as those underlying taste or aesthetics—from noisy ordinal data like triplet comparisons. The authors propose LORE, a framework that jointly optimizes low-rank ordinal embeddings and their intrinsic dimensionality via non-convex Schatten-p quasi-norm regularization, eliminating the need to pre-specify the embedding dimension. An iteratively reweighted algorithm with convergence guarantees is employed for optimization. LORE is the first method to automatically and simultaneously infer both the embedding and its dimensionality directly from ordinal data. Evaluations on synthetic datasets, simulated perceptual spaces, and real-world crowdsourced experiments demonstrate that LORE yields compact, highly accurate, and interpretable low-dimensional representations, substantially improving data efficiency and geometric fidelity.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Knowledge Representation and Reasoning: Preferences
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• intrinsic dimensionality
• ordinal data
• ordinal embedding
• perceptual spaces
• relative similarity

Innovation

• ordinal embedding
• intrinsic dimensionality
• Schatten-p norm
• low-rank learning
• triplet comparisons