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Article Dans Une Revue Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques Année : 2020

On principal curves with a length constraint

Résumé

Principal curves are defined as parametric curves passing through the ``middle'' of a probability distribution in R^d. In addition to the original definition based on self-consistency, several points of view have been considered among which a least square type constrained minimization problem. In this paper, we are interested in theoretical properties satisfied by a constrained principal curve associated to a probability distribution with second-order moment. We study open and closed principal curves f:[0,1]-->R^d with length at most L and show in particular that they have finite curvature whenever the probability distribution is not supported on the range of a curve with length L. We derive from the order 1 condition, expressing that a curve is a critical point for the criterion, an equation involving the curve, its curvature, as well as a random variable playing the role of the curve parameter. This equation allows to show that a constrained principal curve in dimension 2 has no multiple point.
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Dates et versions

hal-01555892 , version 1 (04-07-2017)
hal-01555892 , version 2 (13-10-2019)

Identifiants

Citer

Sylvain Delattre, Aurélie Fischer. On principal curves with a length constraint. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2020, 56 (3), ⟨10.1214/19-AIHP1030⟩. ⟨hal-01555892v2⟩
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