# Metric Learning from Limited Pairwise Preference Comparisons

arXiv: https://arxiv.org/abs/2403.19629 · HTML: https://mlsanity.com/paper/2403.19629/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 47 references that are in the map.
- First posted on arXiv 28 Mar 2024.

## Builds on (most cited first, 10 of 10)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Norm](https://mlsanity.com/paper/1902.03736/) (2019)
- [Divide and Conquer the Embedding Space for Metric Learning](https://mlsanity.com/paper/1906.05990/) (2019)
- [Finite Sample Prediction and Recovery Bounds for Ordinal Embedding](https://mlsanity.com/paper/1606.07081/) (2016)
- [Simultaneous Preference and Metric Learning from Paired Comparisons](https://mlsanity.com/paper/2009.02302/) (2020)
- [Learning Low-Dimensional Metrics](https://mlsanity.com/paper/1709.06171/) (2017)
- [Sample complexity of learning Mahalanobis distance metrics](https://mlsanity.com/paper/1505.02729/) (2015)
- [As you like it: Localization via paired comparisons](https://mlsanity.com/paper/1802.10489/) (2018)
- [One for All: Simultaneous Metric and Preference Learning over Multiple Users](https://mlsanity.com/paper/2207.03609/) (2022)

## Built on by (most cited first, 3 of 3)

- [PAL: Pluralistic Alignment Framework for Learning from Heterogeneous Preferences](https://mlsanity.com/paper/2406.08469/) (2024)
- [Distortion of AI Alignment: Does Preference Optimization Optimize for Preferences?](https://mlsanity.com/paper/2505.23749/) (2025)
- [Metric Learning in an RKHS](https://mlsanity.com/paper/2508.04476/) (2025)

## Abstract (arXiv)

We study metric learning from preference comparisons under the ideal point model, in which a user prefers an item over another if it is closer to their latent ideal item. These items are embedded into ℝd equipped with an unknown Mahalanobis distance shared across users. While recent work shows that it is possible to simultaneously recover the metric and ideal items given O(d) pairwise comparisons per user, in practice we often have a limited budget of o(d) comparisons. We study whether the metric can still be recovered, even though it is known that learning individual ideal items is now no longer possible. We show that in general, o(d) comparisons reveal no information about the metric, even with infinitely many users. However, when comparisons are made over items that exhibit low-dimensional structure, each user can contribute to learning the metric restricted to a low-dimensional subspace so that the metric can be jointly identified. We present a divide-and-conquer approach that achieves this, and provide theoretical recovery guarantees and empirical validation.