Foundations of Computational Mathematics

, Volume 19, Issue 5, pp 991–1011 | Cite as

Interpolation, the Rudimentary Geometry of Spaces of Lipschitz Functions, and Geometric Complexity

  • Shmuel WeinbergerEmail author


We consider seriously the analogy between interpolation of nonlinear functions and manifold learning from samples, and examine the results of transferring ideas from each of these domains to the other. Illustrative examples are given in approximation theory, variational calculus (closed geodesics), and quantitative cobordism theory.


Interpolation Function space Persistent homology Homotopy Quantitative cobordism Embedding 

Mathematics Subject Classification

Primary 68U05 57R75 41A10 Secondary 55Q05 68T10 



  1. 1.
    M. Atiyah and R. Bott. A Lefschetz fixed point formula for elliptic complexes: II. Applications. Ann. of Math., 88:451–491, 1968.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    N. Amenta and M. Bern. Surface reconstruction by voronoi filtering. Discrete Comput. Geom., 22:481–504, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Y. Barzdin. On the realization of networks in three-dimensional space. In A. N. Shiryayev, editor, Selected Works of A. N. Kolmogorov, Volume III, volume 27 of Mathematics and its Applications (Soviet Series), pages 194–202. Springer, Dordrecht, 1993.Google Scholar
  4. 4.
    J. Boissonnat, F. Chazal, and M. Yvinec. Computational geometry and topology for data analysis. To appear.Google Scholar
  5. 5.
    J. Boissonnat, L. Guibas, and S. Oudot. Manifold reconstruction in arbitrary dimensions using witness complexes. Discrete Comput. Geom., 42(1):37–70, 2009.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    S. Buoncristiano and D. Hacon. An elementary geometric proof of two theorems of Thom. Topology, 20(1):97–99, 1981.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    J. Bourgain. On Lipschitz embedding of finite metric spaces in Hilbert space. Israel J. Math., 52:46–52, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    E. Brown. Finite computability of Postnikov complexes. Ann. of Math., 65(1):1–20, 1957.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    W. Browder. Surgery on simply connected manifolds. Springer Verlag, 1972.Google Scholar
  10. 10.
    J. Block and S. Weinberger. Large scale homology theories and geometry. In Geometric Topology: 1993 Georgia International Topology Conference, AMS/IP Stud. Adv. Math., pages 522–569, Providence, RI, 1997. Amer. Math. Soc.Google Scholar
  11. 11.
    G. Chambers, D. Dotterer, F. Manin, and S. Weinberger. Quantitative null-cobordism. J. Amer. Math. Soc., 31(4):1165–1203, 2018.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    S. Cheng, T. Dey, and E. A. Ramos. Manifold reconstruction from point samples. In SODA ’05 Proceedings of the sixtieth annual ACM-SIAM symposium on discrete algorithms, pages 1018–1027, Philadelphia, PA, 2005. Society for Industrial and Applied Mathematics.Google Scholar
  13. 13.
    J. Cheeger and M. Gromov. On the characteristic numbers of complete manifolds of bounded curvature and finite volume. In I. Chavel and H. M. Farkas, editors, Differential Geometry and Complex Analysis, pages 115–154. Springer, Berlin, Heidelberg, 1985.CrossRefGoogle Scholar
  14. 14.
    J. Cha. A topological approach to Cheeger-Gromov universal bounds for von Neumann \(\rho \)-invariants. Comm. Pure and Applied Math., 69:1154–1209, 2016.MathSciNetzbMATHGoogle Scholar
  15. 15.
    G. Carlsson and F. Memoli. Classifying clustering schemes. Found. Comput. Math., 13:221–252, 2013.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    G. Chambers, F. Manin, and S. Weinberger. Quantitative null homotopy and rational homotopy type. Geom. Funct. Anal., 28(3):563–588, 2018.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete Comput. Geom., 37(1):103–120, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    G. Carlsson and A. Zomorodian. Computing persistent homology. Discrete Comput. Geom., 33(2):249–274, 2005.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    F. Cucker and D. Zhou. Learning Theory: An Approximation Theory Viewpoint. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 2007.Google Scholar
  20. 20.
    A. Dranishnikov, S. Ferry, and S. Weinberger. Large Riemannian manifolds which are flexible. Ann. of Math., 157:919–938, 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    A. Dranishnikov, S. Ferry, and S. Weinberger. An infinite dimensional phenomenon in finite dimensional topology. Preprint, 2017.Google Scholar
  22. 22.
    M. DoCarmo. Riemannian Geometry. Birkhäuser Verlag, 1992.Google Scholar
  23. 23.
    H. Edelsbrunner and D. Grayson. Edgewise subdivision of a simplex. Discrete Comput. Geom., 24(4):707–719, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete Comput. Geom., 28:511–533, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Y. Eliashberg and N. Mishachev. Introduction to the h-Principle, volume 48 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002.Google Scholar
  26. 26.
    C. Fefferman. Whitney’s extension problem for \(c^m\). Ann. of Math., 164:313–359, 2006.MathSciNetzbMATHGoogle Scholar
  27. 27.
    C. Fefferman. \(c^m\)-extension by linear operators. Ann. of Math., 166:779–835, 2007.MathSciNetzbMATHGoogle Scholar
  28. 28.
    S. Ferry. Topological finiteness theorems for manifolds in Gromov-Hausdorff space. Duke Math. J., 74(1):95–106, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    C. Fefferman, S. Mitter, and H. Naryanan. Testing the manifold hypothesis. J. Amer. Math. Soc., 29:983–1049, 2016.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    S. Ferry and S. Weinberger. Quantitative algebraic topology and Lipschitz homotopy. Proc. Natl. Acad. Sci. U.S.A., 110:19246–19250, 2013.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    M. Gromov and L. Guth. Generalizations of the Kolmogorov-Barzdin embedding estimates. Duke Math. J., 161:2549–2603, 2012.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    M. Gromov. Homotopical effects of dilation. J. Diff. Geo., 13:313–310, 1978.Google Scholar
  33. 33.
    M. Gromov. Groups of polynomial growth and expanding maps. Publ. Math. IHÉS, 53:51–78, 1981.CrossRefGoogle Scholar
  34. 34.
    M. Gromov. Partial Differential Relations. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer Verlag, 1986.Google Scholar
  35. 35.
    M. Gromov. Dimension, non-linear spectra, and width. In J. Lindenstrauss and V. D. Milman, editors, Geometric Aspects of Functional Analysis, volume 1317 of Lecture Notes in Mathematics, pages 132–184. Springer, Berlin, Heidelberg, 1988.Google Scholar
  36. 36.
    M. Gromov. Metric Structures for Riemannian and non-Riemannian Spaces. Modern Birkhäuser Classics. Birkhäuser Verlag, 1999.Google Scholar
  37. 37.
    M. Gromov. Quantitative homotopy theory. In H. Rossi, editor, Prospects in Mathematics (Princeton, NJ, 1996), pages 45–49. Amer. Math. Soc., Providence, RI, 1999.Google Scholar
  38. 38.
    J. Kleinberg. An impossibility theorm for clustering. In S. Becker, K. Obermayer, and S. Thrun, editors, Advances in Neural Information Processing Systems 15 (NIPS 2002), pages 463–470. MIT Press, Cambridge, MA, 2002.Google Scholar
  39. 39.
    F. Manin. Plato’s cave and differential forms. Preprint, 2018.Google Scholar
  40. 40.
    J. Matousek. Lecture notes on metric embeddings. Preprint, 2013.Google Scholar
  41. 41.
    J. Milnor. Morse Theory, volume 51 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1963.Google Scholar
  42. 42.
    J. Milnor. A note on curvature and fundamental group. J. Differ. Geom., 2:1–7, 1968.MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    F. Manin and S. Weinberger. The Gromov-Guth embedding theorem. Appendix to [11].Google Scholar
  44. 44.
    A. Nabutovsky. Non-recursive functions, knots “with thick ropes” and self-clenching “thick” hyperspheres. Commun. Pure Appl. Math., 48(4):1–50, 1995.MathSciNetzbMATHGoogle Scholar
  45. 45.
    A. Nabutovsky. Morse landscapes of Riemannian functionals and related problems. In Proceedings of the International Congress of Mathematicians: Hyderabad, India, pages 862–881, 2010.Google Scholar
  46. 46.
    P. Niyogi, S. Smale, and S. Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom., 39:419–441, 2008.MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    A. Nabutovsky and S. Weinberger. Variational problems for Riemannian functionals and arithmetic groups. Publ. Math. IHÉS, 92:5–62, 2000.MathSciNetzbMATHGoogle Scholar
  48. 48.
    S. Oudot. Persistence Theory: From Quiver Representations to Data Analysis, volume 209 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2015.Google Scholar
  49. 49.
    I. Polterovich, L. Polterovich, and V. Stojisavljević. Persistence barcodes and Laplace eigenfunctions on surfaces. Preprint, 2017.Google Scholar
  50. 50.
    L. Polterovich, D. Rosen, K. Samvelyan, and J. Zhang. Persistent homology for symplectic topologists. Preprint, 2018.Google Scholar
  51. 51.
    V. Robins. Toward computing homology from finite approximations. Topology Proceedings, 24:503–532, 1999.MathSciNetzbMATHGoogle Scholar
  52. 52.
    J. Roe. Index theory, coarse geometry, and topology of manifolds, volume 90 of CMBS Regional Conference Series in Mathematics. American Mathematical Society, Providence, RI, 1996.Google Scholar
  53. 53.
    S. Smale. The classification of immersions of spheres in Euclidean spaces. Ann. of Math., 69:327–344, 1959.MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    E. Spanier. Algebraic Topology. McGraw-Hill, 1956.Google Scholar
  55. 55.
    R. Strong. Notes on cobordism theory. Mathematical Notes. Princeton University Press, Princeton, NJ, 1968.Google Scholar
  56. 56.
    D. Sullivan. Infinitesimal computations in topology. Publ. Math. IHÉS, 47:269–331, 1977.MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    R. Thom. Quelques propriétés globales des variétés différentiables. Comment. Math. Helv., 28:17–86, 1954.MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    J. Traub and A. Werschulz. Complexity and information. Lezioni Lincee. Cambridge University Press, Cambridge, United Kingdom, 1999.Google Scholar
  59. 59.
    L. Valiant. A theory of the learnable. Commun. ACM, 27(11):1134–1142, 1984.zbMATHCrossRefGoogle Scholar
  60. 60.
    C. T. C. Wall. Surgery on Compact Manifolds. London Mathematical Society Monographs. Academic Press, Cambridge, MA, 1969.Google Scholar
  61. 61.
    S. Weinberger. Computers, Rigidity, and Moduli. Princeton University Press, Princeton, NJ, 2004.Google Scholar
  62. 62.
    S. Weinberger. What is... persistent homology. Notices Amer. Math. Soc., 58(1), 2011.Google Scholar
  63. 63.
    Y. Yomdin and G. Comte. Tame Geometry with Applications in Smooth Analysis, volume 1834 of Lecture Notes in Mathematics. Springer Verlag Berlin Heidelberg, 2004.Google Scholar

Copyright information

© SFoCM 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA

Personalised recommendations