Average-distance problem for parameterized curves

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA
xinyang@andrew.cmu.edu; slepcev@math.cmu.edu

Received: 15 October 2014
Revised: 8 February 2015

Abstract

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

$\mathit{E}\mathrm{\left(}\mathit{\gamma }\mathrm{\right)}\mathrm{=}{\mathrm{\int }}_{R^{\mathit{d}}}\mathit{d}\mathrm{\left(}\mathit{x,}{\mathrm{\Gamma }}_{\mathit{\gamma }}{\mathrm{)}}^{\mathit{p}}\mathrm{d}\mathit{\mu }\mathrm{\right(}\mathit{x}\mathrm{)}\mathrm{+}\mathit{\lambda }\hspace{0.17em}\mathrm{Length}\mathrm{(}\mathit{\gamma }\mathrm{)}$

where γ:I → ℝ^d, I is an interval in ℝ, Γ_γ = γ(I), and d(x,Γ_γ) is the distance of x to Γ_γ. The problem is closely related to the average-distance problem, where the admissible class are the connected sets of finite Hausdorff measure ℋ¹, 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 μ supported in two dimensions the minimizing curve is injective if p ≥ 2 or if μ 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.