Rates of convergence for the continuum limit of nondominated sorting

Cook, Brendan; Calder, Jeff

doi:10.48550/arxiv.2006.05953

Cited by 2 publications

(3 citation statements)

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“…Fundamental questions about whether depths characterize their distributions, and whether depth functions are smooth and well approximated by empirical approximations, are still of current interest [36,40,41]. Similar questions in the context of other depths have also been the topic of recent work [9,13]. Some extensions of the halfspace depth to non-Euclidean settings have also been given in [10], [49].…”

Section: Related Literaturementioning

confidence: 99%

Eikonal depth: an optimal control approach to statistical depths

Molina-Fructuoso¹,

Murray²

2022

Preprint

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Statistical depths provide a fundamental generalization of quantiles and medians to data in higher dimensions. This paper proposes a new type of globally defined statistical depth, based upon control theory and eikonal equations, which measures the smallest amount of probability density that has to be passed through in a path to points outside the support of the distribution: for example spatial infinity. This depth is easy to interpret and compute, expressively captures multi-modal behavior, and extends naturally to data that is non-Euclidean. We prove various properties of this depth, and provide discussion of computational considerations. In particular, we demonstrate that this notion of depth is robust under an aproximate isometrically constrained adversarial model, a property which is not enjoyed by the Tukey depth. Finally we give some illustrative examples in the context of two-dimensional mixture models and MNIST.

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Section: Related Literaturementioning

confidence: 99%

Eikonal depth: an optimal control approach to statistical depths

Molina-Fructuoso¹,

Murray²

2022

Preprint

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“…Calder-Esedoglu-Hero showed that if A consists of randomly scattered points, then, in the large sample limit, h A converges to the solution of a Hamilton-Jacobi equation [CEH14a,CEH14b,CEH15,Cal17]. A rate of convergence was recently establish by Cook-Calder [CC20]. These analyses relied on an equivalence between nondominated sorting of a set A and the longest chain in A, a longest, totally ordered subset of A, l…”

Section: A Generalization Of Nondominated Sortingmentioning

confidence: 99%

“…Calder-Esedoglu-Hero showed that nondominated sorting of random points has a scaling limit described by an explicit Hamilton-Jacobi equation [CEH14a,CEH14b,CEH15,Cal17]. Recently Calder-Cook established a rate of convergence to this continuum limit [CC20] it would be interesting to adapt those ideas to Pareto peeling. 1.3.…”

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confidence: 99%

Hamilton-Jacobi scaling limits of Pareto peeling in 2D

Bou-Rabee¹,

Morfe²

2021

Preprint

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Pareto hull peeling is a discrete algorithm, generalizing convex hull peeling, for sorting points in Euclidean space. We prove that Pareto peeling of a random point set in two dimensions has a scaling limit described by a first-order Hamilton-Jacobi equation and give an explicit formula for the limiting Hamiltonian, which is both non-coercive and non-convex. This contrasts with convex peeling, which converges to curvature flow. The proof involves direct geometric manipulations in the same spirit as Calder (2016).

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Rates of convergence for the continuum limit of nondominated sorting

Cited by 2 publications

References 0 publications

Eikonal depth: an optimal control approach to statistical depths

Eikonal depth: an optimal control approach to statistical depths

Hamilton-Jacobi scaling limits of Pareto peeling in 2D

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