As you like it: Localization via paired comparisons

TL;DR

This paper investigates the problem of estimating a vector from binary comparisons indicating which of two points is closer, providing theoretical bounds, stability results, and adaptive strategies for improved estimation in noisy and randomized settings.

Contribution

It introduces theoretical bounds, stability analysis, and adaptive methods for estimating vectors from paired comparison data, applicable to various preference modeling problems.

Findings

Theoretical bounds for estimation accuracy under randomized models.

Stable embeddings of the target space from a sufficient number of comparisons.

Adaptive comparison strategies significantly improve estimation performance.

Abstract

Suppose that we wish to estimate a vector $\mathbf{x}$ from a set of binary paired comparisons of the form "$\mathbf{x}$ is closer to $\mathbf{p}$ than to $\mathbf{q}$" for various choices of vectors $\mathbf{p}$ and $\mathbf{q}$. The problem of estimating $\mathbf{x}$ 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 $\mathbf{x}$ under a randomized model for $\mathbf{p}$ and $\mathbf{q}$. 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 $\mathbf{p}$ and $\mathbf{q}$, 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 $\mathbf{p}$ and $\mathbf{q}$.