Skip to main content
SpringerLink
Log in

The total variation flow in metric random walk spaces

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

In this paper we study the total variation flow (TVF) in metric random walk spaces, which unifies into a broad framework the TVF on locally finite weighted connected graphs, the TVF determined by finite Markov chains and some nonlocal evolution problems. Once the existence and uniqueness of solutions of the TVF has been proved, we study the asymptotic behaviour of those solutions and, with that aim in view, we establish some inequalities of Poincaré type. In particular, for finite weighted connected graphs, we show that the solutions reach the average of the initial data in finite time. Furthermore, we introduce the concepts of perimeter and mean curvature for subsets of a metric random walk space and we study the relation between isoperimetric inequalities and Sobolev inequalities. Moreover, we introduce the concepts of Cheeger and calibrable sets in metric random walk spaces and characterize calibrability by using the 1-Laplacian operator. Finally, we study the eigenvalue problem whereby we give a method to solve the optimal Cheeger cut problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Alter, F., Caselles, V., Chambolle, A.: A characterization of convex calibrable sets in \({\mathbb{R}}^N\). Math. Ann. 332, 329–366 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L., Di Marino, S.: Equivalent definitions of BV spaces and total variation on metric measures spaces. J. Funct. Anal. 266, 4150–4188 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Clarendon Press, Oxford (2000)

    MATH  Google Scholar 

  4. Ambrosio, L., Pinamonti, A., Speight, G.: Tensorization of Cheeger energies, the space \(H^{1,1}\) and the area formula for graphs. Adv. Math. 281, 1145–1177 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Andreu, F., Caselles, V., Mazón, J.M.: Parabolic Quasilinear Equations Minimizing Linear Growth Functionals. Progress in Mathematics, vol. 223. Birkhauser, Basel (2004)

    Book  MATH  Google Scholar 

  6. Andreu-Vaillo, F., Mazón, J.M., Rossi, J.D., Toledo, J.: Nonlocal Diffusion Problems. Mathematical Surveys and Monographs, vol. 165. AMS, Providence (2010)

    Book  MATH  Google Scholar 

  7. Banerjee, A., Jost, J.: On the spectrum of the normalized graph Laplacian. Linear Algebra Appl. 428, 3015–3022 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bougleux, S., Elmoataz, A., Melkemi, M.: Local and nonlocal discrete regularization on weighted graphs for image and mesh filtering. Int. J. Comput. Vis. 84, 220–236 (2009)

    Article  Google Scholar 

  9. Bénilan, P., Crandall, M.G.: Completely accretive operators. In: Clement, P., et al. (eds.) Semigroups Theory and Evolution Equations (Delft, 1989). Lecture Notes in Pure and Applied Mathematics, vol. 135, pp. 41–75. Marcel Dekker, New York (1991)

    Google Scholar 

  10. Brezis, H.: Operateurs Maximaux Monotones. North Holland, Amsterdam (1973)

    MATH  Google Scholar 

  11. Bruck Jr., R.E.: Asymptotic convergence of nonlinear contraction semigroups in Hilbert spaces. J. Funct. Anal. 18, 15–26 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  12. Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4, 490–530 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bühler, T., Hein, M.: Spectral Clustering Based on the Graph p-Laplacian. In: Proceedings of the 26th International Conference on Machine Learning, pp. 81–88. Omnipress (2009)

  14. Chang, K.C.: Spectrum of the \(1\)-Laplacian and Cheeger’s constant on graphs. J. Graph Theory 81, 167–207 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chang, K.C., Shao, S., Zhang, D.: The 1-Laplacian Cheeger cut: theory and algorithms. J. Comput. Math. 33, 443–467 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chang, K.C., Shao, S., Zhang, D.: Cheeger’s cut, maxcut and the spectral theory of \(1\)-Laplacian on graphs. Sci. China Math. 60, 1963–1980 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Gunning, R.C. (ed.) Problems in Analysis, pp. 195–199. Princeton University Pres, Princeton (1970)

    Google Scholar 

  18. Chung, F.: Spectral Graph Theory (CBMS Regional Conference Series in Mathematics, No. 92). American Mathematical Society, Providence (1997)

    Google Scholar 

  19. Chung, F., Yau, S.T.: Eigenvalues of graphs and Sobolev inequalities. Comb. Probab. Comput. 4, 11–25 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Coulhon, T.: Espaces de Lipschitz et inégalités de Poincaré. J. Funct. Anal. 136, 81–113 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Coulhon, T., Grigor’yan, A., Levin, D.: On isoperimetric profiles of product spaces. Commun. Anal. Geom. 11, 85–120 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Crandall, M.G., Liggett, T.M.: Generation of semigroups of nonlinear transformations on general Banach spaces. Am. J. Math. 93, 265–298 (1971)

    Article  MATH  Google Scholar 

  23. Dávila, J.: On an open question about functions of bounded variation. Calc. Var. Partial Differ. Equ. 15, 519–527 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Elmoataz, A., Lezoray, O., Bougleux, S.: Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing. IEEE Trans. Image Process. 17, 1047–1060 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fridman, V., Kawohl, B.: Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant. Comment. Math. Univ. Carol. 44, 659–667 (2003)

    MathSciNet  MATH  Google Scholar 

  26. García-Trillo, N., Slepčev, D.: Continuum limit of total variation on point clouds. Arch. Ration. Mech. Anal. Res. 220, 193–241 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. García-Trillo, N., Slepčev, D., von Brecht, J., Laurent, T., Bresson, X.: Consistency of Cheeger and ratio graph cuts. J. Mach. Learn. Res. 17, 1–46 (2016)

    MathSciNet  MATH  Google Scholar 

  28. Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Models. Simul. 7, 1005–1028 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hein, M., Bühler, T.: An inverse power method for nonlinear eigenproblems with applications in \(1\)-spectral clustering and sparse PCA. Adv. Neural Inf. Process. Syst. 23, 847–855 (2010)

    Google Scholar 

  30. Hernández-Lerma, O., Laserre, J.B.: Markov Chains and Invariant Probabilities. Birkhäuser Verlag, Basel (2003)

    Book  Google Scholar 

  31. Kindermann, S., Osher, S., Jones, P.: Deblurring and denoising of images by nonlocal functionals. Multiscale Model. Simul. 4, 1091–1115 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lozes, F., Elmoataz, A., Lézoray, O.: Partial difference operators on weighted graphs for image processing on surfaces and point clouds. IEEE Trans. Image Process. 23, 3896–3909 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Marola, N., Miranda Jr., M., Shanmugalingam, N.: Characterizations of sets of finite perimeter using heat kernels in metric spaces. Potential Anal. 45, 609–633 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mazón, J.M., Rossi, J.D., Toledo, J.: Nonlocal perimeter, curvature and minimal surfaces for measurable sets. J. Anal. Math. 138, 235–279 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Mazón, J.M., Rossi, J.D., Toledo, J.: Nonlocal Perimeter, Curvature and Minimal Surfaces for Measurable Sets. Frontiers in Mathematics. Birkhäuser, Basel (2019)

    Book  MATH  Google Scholar 

  36. Mazón, J.M., Solera, M., Toledo, J.: The heat flow on metric random walk spaces. J. Math. Anal. Appl. 483, 123645 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mazón, J. M., Solera, M., Toledo, J.: \((BV,L^p)\)-decomposition, \(p=1,2\), of functions in metric random walk spaces. arXiv:1907.10650

  38. Maz’ya, V.: Sobolev Spaces. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  39. Maz’ya, V.: Lectures on Isoperimetric and Isocapacitary Inequalities in the theory of Sobolev spaces. Contemp. Math. 338, 307–340 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University lecture Series, 22. American Mathematical Society, Providence (2001)

    Google Scholar 

  41. Miranda, M.Jr: Functions of bounded variation on good metric spaces. J. Math. Pures Appl. 82, 975–1004 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  42. Monti, R.: Rearrangements in metric spaces and in the Heisenberg group. J. Geom. Anal. 24, 1673–1715 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Norris, J.R.: Markov Chains. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  44. Ollivier, Y.: Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256, 810–864 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algoritms. Physica D 60, 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  46. Saloff-Coste, L.: Aspects of Sobolev-Type Inequalities. London Mathematical Society Lecture Notes Series, vol. 289. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  47. Szlam, A., Bresson, X.: Total variation and Cheeger cuts. In: Proceedings of the 27th International Conference on Machine Learning, Haifa, Israel (2010)

  48. Tillich, J.-P.: Edge isoperimetric inequalities for product graphs. Discrete Math. 213, 291–320 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  49. Varopoulos, N.T.: Isoperimetric inequalities and Marrkov chains. J. Funct. Anal. 63, 215–239 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  50. van Gennip, Y., Guillen, N., Osting, B., Bertozzi, A.: Mean curvature, threshold dynamics, and phase field theory on finite graphs. Milan J. Math. 82, 3–65 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  51. von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17, 395–416 (2007)

    Article  MathSciNet  Google Scholar 

  52. Yaroslavsky, L.P.: Digital Picture Processing. An Introduction. Springer, Berlin (1985)

    Book  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the anonymous referee whose comments after a detailed reading of the paper allowed them to improve its presentation. The authors have been partially supported by the Spanish MICIU and FEDER, Project PGC2018-094775-B-100. The second author was also supported by the Spanish MICIU under Grant BES-2016-079019, which is also supported by the European FSE.

Author information

Authors and Affiliations

  1. Departament d’Anàlisi Matemàtica, Universitat de València, Dr. Moliner 50, 46100, Burjassot, Spain

    José M. Mazón, Marcos Solera & Julián Toledo

Authors
  1. José M. Mazón
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcos Solera
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Julián Toledo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to José M. Mazón.

Additional information

Communicated by Y. Giga.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mazón, J.M., Solera, M. & Toledo, J. The total variation flow in metric random walk spaces. Calc. Var. 59, 29 (2020). https://doi.org/10.1007/s00526-019-1684-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-019-1684-z

Mathematics Subject Classification

Search

Navigation