# Learning Low-Dimensional Metrics

arXiv: https://arxiv.org/abs/1709.06171 · HTML: https://mlsanity.com/paper/1709.06171/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Sep 2017.

## Built on by (most cited first, 9 of 9)

- [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)
- [Self-Supervised Metric Learning in Multi-View Data: A Downstream Task Perspective](https://mlsanity.com/paper/2106.07138/) (2021)
- [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)
- [Nearest Neighbor Search Under Uncertainty](https://mlsanity.com/paper/2103.05057/) (2021)
- [Representer Theorems for Metric and Preference Learning: Geometric Insights and Algorithms](https://mlsanity.com/paper/2304.03720/) (2023)
- [Dictionary Learning: The Complexity of Learning Sparse Superposed Features with Feedback](https://mlsanity.com/paper/2502.05407/) (2025)
- [Metric Learning in an RKHS](https://mlsanity.com/paper/2508.04476/) (2025)

## Abstract (arXiv)

This paper investigates the theoretical foundations of metric learning, focused on three key questions that are not fully addressed in prior work: 1) we consider learning general low-dimensional (low-rank) metrics as well as sparse metrics; 2) we develop upper and lower (minimax)bounds on the generalization error; 3) we quantify the sample complexity of metric learning in terms of the dimension of the feature space and the dimension/rank of the underlying metric;4) we also bound the accuracy of the learned metric relative to the underlying true generative metric. All the results involve novel mathematical approaches to the metric learning problem, and lso shed new light on the special case of ordinal embedding (aka non-metric multidimensional scaling).