Free Access
Issue
ESAIM: COCV
Volume 27, 2021
Article Number 8
Number of page(s) 25
DOI https://doi.org/10.1051/cocv/2021002
Published online 03 March 2021
  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). [Google Scholar]
  2. G. Biau and A. Fischer, Parameter selection for principal curves. IEEE Trans. Inf. Theory 58 (2011) 1924–1939. [Google Scholar]
  3. E. Bretin, J.-O. Lachaud and É. Oudet, Regularization of discrete contour by Willmore energy. J. Math. Imag. Vision 40 (2011) 214–229. [Google Scholar]
  4. G. Buttazzo, E. Mainini and E. Stepanov, Stationary configurations for the average distance functional and related problems. Control Cybernet. 38 (2009) 1107–1130. [Google Scholar]
  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. [Google Scholar]
  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). [Google Scholar]
  7. G. Buttazzo, A. Pratelli and E. Stepanov, Optimal pricing policies for public transportation networks. SIAM J. Optim. 16 (2006) 826–853. [Google Scholar]
  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). [Google Scholar]
  9. G. Buttazzo and F. Santambrogio, A mass transportation model for the optimal planning of an urban region. SIAM Rev. 51 (2009) 593–610. [Google Scholar]
  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. [Google Scholar]
  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. [Google Scholar]
  12. C. de Boor A practical guide to splines. Vol. 27 of Applied Mathematical Sciences. Springer-Verlag. New York, revised ed. (2001). [Google Scholar]
  13. S. Delattre and A. Fischer, On principal curves with a length constraint. Ann. Inst. Henri Poincaré Probab. Statist. 56 (2020) 2108–2140. [Google Scholar]
  14. P. Delicado, Another look at principal curves and surfaces. J. Multivariate Anal. 77 (2001) 84–116. [Google Scholar]
  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. [Google Scholar]
  16. Q. Du, C. Liu, R. Ryham and X. Wang, A phase field formulation of the Willmore problem. Nonlinearity 18 (2005) 1249–1267. [Google Scholar]
  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. [Google Scholar]
  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). [Google Scholar]
  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. [Google Scholar]
  20. S. Gerber and R. Whitaker, Regularization-free principal curve estimation. J. Machine Learning Res. 14 (2013) 1285–1302. [Google Scholar]
  21. V.G.A. Goss, Snap buckling, writhing and loop formation in twisted rods. Ph.D. thesis, University College London (2003). [Google Scholar]
  22. T. Hastie and W. Stuetzle, Principal curves. J. Am. Statist. Assoc. 84 (1989) 502–516. [Google Scholar]
  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. [Google Scholar]
  24. S. Kirov and D. Slepčev, Multiple penalized principal curves: analysis and computation. J. Math. Imaging Vision 59 (2017) 234–256. [Google Scholar]
  25. A. Lemenant, About the regularity of average distance minimizers in ℝ2. J. Convex Anal. 18 (2011) 949–981. [Google Scholar]
  26. R. Levien, The elastica: a mathematical history (2008). [Google Scholar]
  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. [Google Scholar]
  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. [Google Scholar]
  29. X.Y. Lu and D. Slepčev, Average-distance problem for parameterized curves. ESAIM: COCV 22 (2016) 404–416. [EDP Sciences] [Google Scholar]
  30. C. Mantegazza and A.C. Mennucci, Hamilton-Jacobi equations and distance functions on Riemannian manifolds. Appl. Math. Optim. 47 (2003) 1–25. [Google Scholar]
  31. U. Ozertem and D. Erdogmus, Locally defined principal curves and surfaces. J. Mach. Learn. Res. 12 (2011) 1249–1286. [Google Scholar]
  32. E. Paolini and E. Stepanov, Qualitative properties of maximum distance minimizers and average distance minimizers in ℝn. J. Math. Sci. (N. Y.) 122 (2004) 3290–3309. [Google Scholar]
  33. P. Polak and G. Wolansky, The lazy travelling salesman problem in ℝ2. ESAIM: COCV 13 (2007) 538–552. [CrossRef] [EDP Sciences] [Google Scholar]
  34. D. Slepčev, Counterexample to regularity in average-distance problem. Ann. Inst. Henri Poincaré Anal. Non Linéaire 31 (2014) 169–184. [Google Scholar]
  35. A.J. Smola, S. Mika, B. Schölkopf and R.C. Williamson, Regularized principal manifolds. J. Mach. Learn. Res. 1 (2001) 179–209. [Google Scholar]
  36. R. Tibshirani, Principal curves revisited. Stat. Comput. 2 (1992) 182–190. [Google Scholar]

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