A Hamilton--Jacobi Equation for the Continuum Limit of Nondominated Sorting

Calder, Jeff; Esedoḡlu, Selim; Hero, Alfred O.

doi:10.1137/13092842x

Cited by 24 publications

(65 citation statements)

References 40 publications

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“…In conclusion, we have presented an overview of our recent work on a continuum limit for non-dominated sorting [35], [36]. We identified a Hamilton-Jacobi partial differential equation (PDE) for this continuum limit, and showed how to numerically solve the PDE efficiently.…”

Section: Discussionmentioning

confidence: 99%

“…In particular, c 1 = 1, c 2 = 2 and c d e as d → ∞. The proof of Theorem 1 will appear in the SIAM Journal on Mathematical Analysis in 2014 [35].…”

Section: B Continuum Limitmentioning

confidence: 92%

“…In this paper, we overview the results of some recent work by the authors on a continuum limit for non-dominated sorting [35], [36]. In particular, we prove that in the (random) large sample size limit, the non-dominated fronts converge almost surely to the level sets of a function that satisfies a Hamilton-Jacobi partial differential equation (PDE).…”

Section: Introductionmentioning

confidence: 89%

“…For example, in [35], we used Theorem 1 to show that non-dominated sorting is asymptotically stable under bounded random perturbations. Furthermore, convexity (or lack thereof) is a crucial property of Pareto fronts [1].…”

Section: B Continuum Limitmentioning

confidence: 99%

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A continuum limit for non-dominated sorting

Calder

Esedoḡlu

Hero

2014

2014 Information Theory and Applications Workshop (ITA)

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Abstract-Non-dominated sorting is an important combinatorial problem in multi-objective optimization, which is ubiquitous in many fields of science and engineering. In this paper, we overview the results of some recent work by the authors on a continuum limit for non-dominated sorting. In particular, we have discovered that in the (random) large sample size limit, the non-dominated fronts converge almost surely to the level sets of a function that satisfies a Hamilton-Jacobi partial differential equation (PDE). We show how this PDE can be used to design a fast, potentially sublinear, approximate non-dominated sorting algorithm, and we show the results of applying the algorithm to real data from an anomaly detection problem.

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Section: Discussionmentioning

confidence: 99%

“…In particular, c 1 = 1, c 2 = 2 and c d e as d → ∞. The proof of Theorem 1 will appear in the SIAM Journal on Mathematical Analysis in 2014 [35].…”

Section: B Continuum Limitmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 89%

Section: B Continuum Limitmentioning

confidence: 99%

See 2 more Smart Citations

A continuum limit for non-dominated sorting

Calder

Esedoḡlu

Hero

2014

2014 Information Theory and Applications Workshop (ITA)

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“…In Section 3, we relate deterministically the number of fronts hitting a given soma to the length of the longest increasing (for some specific order) subsequence of the points (time and space) from which these fronts start. In Section 4, which is very technical, we adapt to our context the result of Calder, Esedoglu and Hero [6]. The proofs of our main results concerning the hard and soft models are handled in Sections 5 and 6.…”

Section: Perspectives One Important Question Remains Open: Does Propmentioning

confidence: 99%

On a toy network of neurons interacting through their dendrites

Fournier¹,

Tanré²,

Veltz³

2020

Ann. Inst. H. Poincaré Probab. Statist.

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Consider a large number n of neurons, each being connected to approximately N other ones, chosen at random. When a neuron spikes, which occurs randomly at some rate depending on its electric potential, its potential is set to a minimum value v min , and this initiates, after a small delay, two fronts on the (linear) dendrites of all the neurons to which it is connected. Fronts move at constant speed. When two fronts (on the dendrite of the same neuron) collide, they annihilate. When a front hits the soma of a neuron, its potential is increased by a small value wn. Between jumps, the potentials of the neurons are assumed to drift in [v min , ∞), according to some well-posed ODE. We prove the existence and uniqueness of a heuristically derived mean-field limit of the system when n, N → ∞ with wn N −1/2 . We make use of some recent versions of the results of Deuschel and Zeitouni [15] concerning the size of the longest increasing subsequence of an i.i.d. collection of points in the plan. We also study, in a very particular case, a slightly different model where the neurons spike when their potential reach some maximum value vmax, and find an explicit formula for the (heuristic) mean-field limit.2010 Mathematics Subject Classification. 60K35, 60J75, 92C20. Key words and phrases. Mean-field limit, Propagation of chaos, nonlinear stochastic differential equations, Ulam's problem, Longest increasing subsequence, Biological neural networks. arXiv:1802.04118v2 [math.PR] 26 Apr 2019 1.2. Biological background. Although the above particle systems are toy models, they are strongly inspired by biology.General organization. A neuron is a specialized cell type of the central nervous system. It is composed of sub-cellular domains which serve different functions, see Kandel [23]. More precisely, the neuron is comprised of a dendrite, a soma (otherwise known as the cell body) and an axon. See Figure 1 for a schematic description. The neurons are connected with synapses which are the interface between the axons and the dendrites. On Figure 1, the axon of the neuron i is connected, through synapses, to the dendrites of the neurons j and k.

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Numerical schemes and rates of convergence for the Hamilton–Jacobi equation continuum limit of nondominated sorting

Calder

2017

Numer. Math.

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Non-dominated sorting arranges a set of points in n-dimensional Euclidean space into layers by repeatedly removing the coordinatewise minimal elements. It was recently shown that nondominated sorting of random points has a Hamilton-Jacobi equation continuum limit. The obvious numerical scheme for this PDE has a slow convergence rate of O(h 1 n ). In this paper, we introduce two new numerical schemes that have formal rates of O(h) and we prove the usual O( √ h) theoretical rates. We also present the results of numerical simulations illustrating the difference between the formal and theoretical rates.

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A Hamilton--Jacobi Equation for the Continuum Limit of Nondominated Sorting

Cited by 24 publications

References 40 publications

A continuum limit for non-dominated sorting

A continuum limit for non-dominated sorting

On a toy network of neurons interacting through their dendrites

Numerical schemes and rates of convergence for the Hamilton–Jacobi equation continuum limit of nondominated sorting

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