On principal curves with a length constraint

TL;DR

This paper investigates the theoretical properties of length-constrained principal curves in R^d, showing they have finite curvature and no multiple points in dimension 2, under certain distribution conditions.

Contribution

It provides a rigorous analysis of constrained principal curves, establishing conditions for finite curvature and uniqueness properties in the plane.

Findings

Principal curves with length constraints have finite curvature.

In dimension 2, such curves have no multiple points.

Derived an equation involving the curve, curvature, and a random parameter.

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.