# Simultaneous Preference and Metric Learning from Paired Comparisons

arXiv: https://arxiv.org/abs/2009.02302 · HTML: https://mlsanity.com/paper/2009.02302/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Sep 2020.

## Built on by (most cited first, 10 of 10)

- [PAL: Pluralistic Alignment Framework for Learning from Heterogeneous Preferences](https://mlsanity.com/paper/2406.08469/) (2024)
- [One for All: Simultaneous Metric and Preference Learning over Multiple Users](https://mlsanity.com/paper/2207.03609/) (2022)
- [Perceptual adjustment queries and an inverted measurement paradigm for low-rank metric learning](https://mlsanity.com/paper/2309.04626/) (2023)
- [Metric Learning from Limited Pairwise Preference Comparisons](https://mlsanity.com/paper/2403.19629/) (2024)
- [PrefGen: Preference Guided Image Generation with Relative Attributes](https://mlsanity.com/paper/2304.00185/) (2023)
- [The Battling Influencers Game: Nash Equilibria Structure of a Potential Game and Implications to Value Alignment](https://mlsanity.com/paper/2502.01127/) (2025)
- [Angular triangle distance for ordinal metric learning](https://mlsanity.com/paper/2211.15200/) (2022)
- [Representer Theorems for Metric and Preference Learning: Geometric Insights and Algorithms](https://mlsanity.com/paper/2304.03720/) (2023)
- [Metric Learning in an RKHS](https://mlsanity.com/paper/2508.04476/) (2025)
- [LLM-Augmented Computational Phenotyping of Long Covid](https://mlsanity.com/paper/2603.18115/) (2026)

## Abstract (arXiv)

A popular model of preference in the context of recommendation systems is the so-called ideal point model. In this model, a user is represented as a vector u together with a collection of items x1, ldots, xN in a common low-dimensional space. The vector u represents the user's "ideal point," or the ideal combination of features that represents a hypothesized most preferred item. The underlying assumption in this model is that a smaller distance between u and an item xj indicates a stronger preference for xj. In the vast majority of the existing work on learning ideal point models, the underlying distance has been assumed to be Euclidean. However, this eliminates any possibility of interactions between features and a user's underlying preferences. In this paper, we consider the problem of learning an ideal point representation of a user's preferences when the distance metric is an unknown Mahalanobis metric. Specifically, we present a novel approach to estimate the user's ideal point u and the Mahalanobis metric from paired comparisons of the form "item xi is preferred to item xj." This can be viewed as a special case of a more general metric learning problem where the location of some points are unknown a priori. We conduct extensive experiments on synthetic and real-world datasets to exhibit the effectiveness of our algorithm.