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Average-distance problem with curvature penalization for data parameterization: regularity of minimizers. (English) Zbl 1467.49040

Summary: We propose a model for finding one-dimensional structure in a given measure. Our approach is based on minimizing an objective functional which combines the average-distance functional to measure the quality of the approximation and penalizes the curvature, similarly to the elastica functional. Introducing the curvature penalization overcomes some of the shortcomings of the average-distance functional, in particular the lack of regularity of minimizers. We establish existence, uniqueness and regularity of minimizers of the proposed functional. In particular we establish \(C^{1,1}\) estimates on the minimizers.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49Q10 Optimization of shapes other than minimal surfaces
35B65 Smoothness and regularity of solutions to PDEs
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[1] J. Bertoulli, Curvatura laminae elasticae. Ejus identitas cum curvatura lintei a pondere inclusi fluidi expansi. Radii circulorum osculantium in terminis simplicissimis exhibiti; una cum novis quibusdarn theorematis huc pertinentibus. Acta Erudirorum (1694).
[2] G. Biau and A. Fischer, Parameter selection for principal curves. IEEE Trans. Inf. Theory 58 (2011) 1924-1939. · Zbl 1365.62262
[3] E. Bretin, J.-O. Lachaud and É. Oudet, Regularization of discrete contour by Willmore energy. J. Math. Imag. Vision 40 (2011) 214-229. · Zbl 1255.68212
[4] G. Buttazzo, E. Mainini and E. Stepanov, Stationary configurations for the average distance functional and related problems. Control Cybernet. 38 (2009) 1107-1130. · Zbl 1239.49029
[5] G. Buttazzo, E. Oudet and E. Stepanov, Optimal transportation problems with free dirichlet regions, in Variational methods for discontinuous structures. Vol. 51 of Progr. Nonlinear Differential Equations Appl. Birkhäuser, Basel (2002) 41-65. · Zbl 1055.49029
[6] G. Buttazzo, A. Pratelli, S. Solimini and E. Stepanov, Optimal urban networks via mass transportation. Vol. 1961 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2009). · Zbl 1190.90003 · doi:10.1007/978-3-540-85799-0
[7] G. Buttazzo, A. Pratelli and E. Stepanov, Optimal pricing policies for public transportation networks. SIAM J. Optim. 16 (2006) 826-853. · Zbl 1093.49030
[8] G. Buttazzo and F. Santambrogio, A model for the optimal planning of an urban area. SIAM J. Math. Anal. 37 (2005) 514-530 (electronic). · Zbl 1109.49027
[9] G. Buttazzo and F. Santambrogio, A mass transportation model for the optimal planning of an urban region. SIAM Rev. 51 (2009) 593-610. · Zbl 1170.49013
[10] G. Buttazzo and E. Stepanov, Optimal transportation networks as free dirichlet regions for the Monge-Kantorovich problem. Ann. Sc. Norm. Super. Pisa Cl. Sci. 2 (2003) 631-678. · Zbl 1127.49031
[11] G. Buttazzo and E. Stepanov, Minimization problems for average distance functionals, in topics from the mathematical heritage of E. de Giorgi. Vol. 14 of Quad. Mat., Dept. Math. Seconda Univ. Napoli, Caserta. Cal. Var. Geom. Measure Theor. 14 (2004) 47-83. · Zbl 1089.49040
[12] C. de Boor A practical guide to splines. Vol. 27 of Applied Mathematical Sciences. Springer-Verlag. New York, revised ed. (2001). · Zbl 0987.65015
[13] S. Delattre and A. Fischer, On principal curves with a length constraint. Ann. Inst. Henri Poincaré Probab. Statist. 56 (2020) 2108-2140. · Zbl 1454.60031
[14] P. Delicado, Another look at principal curves and surfaces. J. Multivariate Anal. 77 (2001) 84-116. · Zbl 1033.62048
[15] P.W. Dondl, L. Mugnai and M. Röger, A phase field model for the optimization of the Willmore energy in the class of connected surfaces. SIAM J. Math. Anal. 46 (2014) 1610-1632. · Zbl 1293.49106
[16] Q. Du, C. Liu, R. Ryham and X. Wang, A phase field formulation of the Willmore problem. Nonlinearity 18 (2005) 1249-1267. · Zbl 1125.35366
[17] T. Duchamp and W. Stuetzle, Geometric properties of principal curves in the plane, in Robust statistics, data analysis, and computerintensive methods (Schloss Thurnau, 1994). Vol. 109 of Lecture Notes in Statist. Springer, New York (1996) 135-152. · Zbl 0839.62040
[18] L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti. Lausannæ, Genevæ, apud Marcum-Michaelem Bousquet & socios (1744). · Zbl 0788.01072
[19] S. Gerber, T. Tasdizen and R. Whitaker, Dimensionality reduction and principal surfaces via kernel map manifolds, in 2009 IEEE 12th International Conference on Computer Vision. IEEE (2009) 529-536.
[20] S. Gerber and R. Whitaker, Regularization-free principal curve estimation. J. Machine Learning Res. 14 (2013) 1285-1302. · Zbl 1317.68159
[21] V.G.A. Goss, Snap buckling, writhing and loop formation in twisted rods. Ph.D. thesis, University College London (2003).
[22] T. Hastie and W. Stuetzle, Principal curves. J. Am. Statist. Assoc. 84 (1989) 502-516. · Zbl 0679.62048
[23] B. Kégl, A. Krzyzak, T. Linder and K. Zeger, Learning and design of principal curves. IEEE Trans. Pattern Anal. Mach. Intell. 22 (2000) 281-297.
[24] S. Kirov and D. Slepčev, Multiple penalized principal curves: analysis and computation. J. Math. Imaging Vision 59 (2017) 234-256. · Zbl 1385.49023
[25] A. Lemenant, About the regularity of average distance minimizers in ℝ^2. J. Convex Anal. 18 (2011) 949-981. · Zbl 1238.49054
[26] R. Levien, The elastica: a mathematical history (2008).
[27] X.Y. Lu, Example of minimizer of the average-distance problem with non closed set of corners. Rendiconti del Seminario Matematico della Università di Padova 137 (2017) 19-55. · Zbl 1368.49055
[28] X.Y. Lu and D. Slepčev, Properties of minimizers of average-distance problem via discrete approximation of measures. SIAM J. Math. Anal. 45 (2013) 3114-3131. · Zbl 1280.49064
[29] X.Y. Lu and D. Slepčev, Average-distance problem for parameterized curves. ESAIM: COCV 22 (2016) 404-416. · Zbl 1338.49094
[30] C. Mantegazza and A.C. Mennucci, Hamilton-Jacobi equations and distance functions on Riemannian manifolds. Appl. Math. Optim. 47 (2003) 1-25. · Zbl 1048.49021
[31] U. Ozertem and D. Erdogmus, Locally defined principal curves and surfaces. J. Mach. Learn. Res. 12 (2011) 1249-1286. · Zbl 1280.62071
[32] E. Paolini and E. Stepanov, Qualitative properties of maximum distance minimizers and average distance minimizers in ℝ^n. J. Math. Sci. (N. Y.) 122 (2004) 3290-3309. · Zbl 1099.49029
[33] P. Polak and G. Wolansky, The lazy travelling salesman problem in ℝ^2. ESAIM: COCV 13 (2007) 538-552. · Zbl 1153.90014 · doi:https://www.esaim-cocv.org/articles/cocv/abs/2007/03/cocv0521/cocv0521.html
[34] D. Slepčev, Counterexample to regularity in average-distance problem. Ann. Inst. Henri Poincaré Anal. Non Linéaire 31 (2014) 169-184. · Zbl 1286.49055
[35] A.J. Smola, S. Mika, B. Schölkopf and R.C. Williamson, Regularized principal manifolds. J. Mach. Learn. Res. 1 (2001) 179-209. · Zbl 1005.68137
[36] R. Tibshirani, Principal curves revisited. Stat. Comput. 2 (1992) 182-190.
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