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

$$E\left(\right)={\int}_{{\mathbb{R}}^{d}}d{(x,{\Gamma}_{\gamma})}^{p}\mathrm{d}\mu \left(x\right)+\lambda \phantom{\rule{0.166667em}{0ex}}\text{Length}\left(\gamma \right)$$ |

where $\gamma :I\to {\mathbb{R}}^{d}$, $I$ is an interval in $\mathbb{R}$, $\Gamma {}_{\gamma}=\gamma \left(I\right)$, 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.

DOI: 10.1051/cocv/2015011

Keywords: Average-distance problem, principal curves, nonlocal variational problems

^{1}; Slepčev, Dejan

^{1}

@article{COCV_2016__22_2_404_0, author = {Lu, Xin Yang and Slep\v{c}ev, Dejan}, title = {Average-distance problem for parameterized curves}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {404--416}, publisher = {EDP-Sciences}, volume = {22}, number = {2}, year = {2016}, doi = {10.1051/cocv/2015011}, mrnumber = {3491776}, zbl = {1338.49094}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2015011/} }

TY - JOUR AU - Lu, Xin Yang AU - Slepčev, Dejan TI - Average-distance problem for parameterized curves JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 404 EP - 416 VL - 22 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015011/ DO - 10.1051/cocv/2015011 LA - en ID - COCV_2016__22_2_404_0 ER -

%0 Journal Article %A Lu, Xin Yang %A Slepčev, Dejan %T Average-distance problem for parameterized curves %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 404-416 %V 22 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015011/ %R 10.1051/cocv/2015011 %G en %F COCV_2016__22_2_404_0

Lu, Xin Yang; Slepčev, Dejan. Average-distance problem for parameterized curves. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 2, pp. 404-416. doi : 10.1051/cocv/2015011. http://archive.numdam.org/articles/10.1051/cocv/2015011/

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