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arXiv · 1802.10489

As you like it: Localization via paired comparisons

Andrew K. Massimino·Mark A. Davenport
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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 $\mathbf{q}$.

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Andrew K. Massimino, Mark A. Davenport. 2018-02-19. As you like it: Localization via paired comparisons. https://arxiv.org/abs/1802.10489

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