# 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.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-01555892
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

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CourbesSDAFrev.pdf
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### Identifiers

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

### Citation

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

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