Publication 14-CNA-024
Average-distance problem for parameterized curves
Xin Yang Lu
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
xinyang@andrew.cmu.edu
Dejan Slepčev
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
slepcev@andrew.cmu.edu
Abstract: We consider approximating a measure by a parameterized curve
subject to length penalization.
That is for a given finite positive compactly supported measure $\mu$,
for $p \geq 1$ and $\lambda>0$ we consider the functional
\[ E(\gamma) = \int_{\mathbb{R}^d} d(x, \Gamma_\gamma)^p d\mu(x) +
\lambda \,\textrm{Length}(\gamma) \]
where $\gamma:I \to \mathbb{R}^d$, $I$ is an interval in $\mathbb{R}$,
$\Gamma_\gamma = \gamma(I)$, 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
$\mathcal H^1$, 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 $p \geq 2$ 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.
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