# As you like it: Localization via paired comparisons

arXiv: https://arxiv.org/abs/1802.10489 · HTML: https://mlsanity.com/paper/1802.10489/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Feb 2018.

## Built on by (most cited first, 6 of 6)

- [One for All: Simultaneous Metric and Preference Learning over Multiple Users](https://mlsanity.com/paper/2207.03609/) (2022)
- [Active Preference Learning for Ordering Items In- and Out-of-sample](https://mlsanity.com/paper/2405.03059/) (2024)
- [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)
- [Source-Free Domain Adaptation for SSVEP-based Brain-Computer Interfaces](https://mlsanity.com/paper/2305.17403/) (2023)
- [Representer Theorems for Metric and Preference Learning: Geometric Insights and Algorithms](https://mlsanity.com/paper/2304.03720/) (2023)

## Abstract (arXiv)

Suppose that we wish to estimate a vector x from a set of binary paired comparisons of the form "x is closer to p than to q" for various choices of vectors p and q. The problem of estimating x from this type of observation arises in a variety of contexts, including nonmetric multidimensional scaling, "unfolding," and ranking problems, often because it provides a powerful and flexible model of preference. We describe theoretical bounds for how well we can expect to estimate x under a randomized model for p and q. We also present results for the case where the comparisons are noisy and subject to some degree of error. Additionally, we show that under a randomized model for p and q, a suitable number of binary paired comparisons yield a stable embedding of the space of target vectors. Finally, we also show that we can achieve significant gains by adaptively changing the distribution for choosing p and q.