Skip to Main content Skip to Navigation
Preprints, Working Papers, ...

On principal curves with a length constraint

Abstract : 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.
Document type :
Preprints, Working Papers, ...
Complete list of metadatas
Contributor : Aurélie Fischer <>
Submitted on : Sunday, October 13, 2019 - 12:36:36 PM
Last modification on : Friday, March 27, 2020 - 4:01:49 AM
Document(s) archivé(s) le : Tuesday, January 14, 2020 - 12:14:51 PM


Files produced by the author(s)


  • HAL Id : hal-01555892, version 2
  • ARXIV : 1707.01326


Sylvain Delattre, Aurélie Fischer. On principal curves with a length constraint. 2017. ⟨hal-01555892v2⟩



Record views


Files downloads