Average-distance problem for parameterized curves

2014/11/11 by Xin Yang Lu, Lu, Xin Yang, Dejan Slepčev +1
Mathematics · #35B65 #49K10 #49Q10 #49Q20 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B65 #msc:49K10 #msc:49Q10 #msc:49Q20

paper · pdf · doi:10.48550/arxiv.1411.2673

arxiv created 2014/11/11 · arxiv updated 2014/11/12

Abstract

We consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite positive compactly supported measure μ, for p ≥ 1 and λ>0 we consider the functional E(γ) = ∫_ℝ^d d(x, Γ_γ)^p dμ(x) + λ \textrmLength(γ) 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 \mathcal 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.

Related

• Average-distance problem with curvature penalization for data parameterization: regularity of minimizers
• The average distance problem with an Euler elastica penalization
• Structure of average distance minimizers in general dimensions
• The average distance problem with perimeter-to-area ratio penalization
• Distance distributions and inverse problems for metric measure spaces