The geometry of random paired comparisons

Massimino, Andrew K.; Davenport, Mark A.

doi:10.1109/icassp.2017.7953383

Cited by 2 publications

(3 citation statements)

References 14 publications

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“…In particular, our results are largely inspired by those in [21,3], and borrow techniques from [17] in the proofs of some of our main results. Note that in this work we extend preliminary results first presented in [23,24].…”

Section: Related Worksupporting

confidence: 86%

“…Letting δ = ζR/ √ n and setting C 1 , c 1 , C 2 , c 1 appropriately 4 this reduces to (24). Lower bounding the probability by 1 − η, we obtain…”

Section: Stable Embeddingmentioning

confidence: 99%

“…In the context of a hypothetical recovery problem, suppose x is a parameter of interest and y is an estimate produced by any algorithm. Then (24) says that if we want to recover x to within error ϵ, the algorithm should look for vectors y which is have up to O(ϵ) incorrect comparisons. Likewise, if a y can be found having up to O(ϵ) comparison errors, we have the same O(ϵ) guarantee on the Euclidean error of the estimate.…”

Section: Stable Embeddingmentioning

confidence: 99%

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As you like it: Localization via paired comparisons

Massimino,

Davenport

2018

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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 that we can achieve significant gains by adaptively changing the distribution for choosing p and q.

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