Average-distance problem for parameterized curves

Lu, Xin Yang; Slepčev, Dejan

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 2, pp. 404-416.

We consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite compactly supported measure $\mu$, with $\mu \left({ℝ}^d\right)>0$ for $p\ge 1$ and $\lambda >0$ we consider the functional

$$E\left(\right)={\int }_{{ℝ}^d}d{(x,{\Gamma }_{\gamma })}^p\mathrm{d}\mu \left(x\right)+\lambda \text{Length}\left(\gamma \right)$$

where $\gamma :I\to {ℝ}^d$, $I$ is an interval in $ℝ$, $\Gamma {}_{\gamma }=\gamma \left(I\right)$, and $d(x,\Gamma {}_{\gamma })$ is the distance of $x$ to $\Gamma {}_{\gamma }$. The problem is closely related to the average-distance problem, where the admissible class are the connected sets of finite Hausdorff measure ${\mathscr{H}}^1$, and to (regularized) principal curves studied in statistics. We obtain regularity of minimizers in the form of estimates on the total curvature of the minimizers. We prove that for measures $\mu$ supported in two dimensions the minimizing curve is injective if $p\ge 2$ or if $\mu$ has bounded density. This establishes that the minimization over parameterized curves is equivalent to minimizing over embedded curves and thus confirms that the problem has a geometric interpretation.

Classification : 49Q20, 49K10, 49Q10, 35B65
Mots clés : Average-distance problem, principal curves, nonlocal variational problems

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