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

TL;DR

LORE is a scalable framework that jointly learns the intrinsic dimensionality and ordinal embedding of subjective perceptual spaces from noisy triplet comparisons, improving interpretability and data efficiency.

Contribution

LORE introduces a novel joint learning approach using Schatten-$p$ quasi norm regularization to automatically recover both the embedding and its dimensionality.

Findings

Learns compact, interpretable low-dimensional embeddings

Achieves high accuracy in recovering latent perceptual geometry

Demonstrates effectiveness on synthetic and real-world 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.