Skip to main content

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

Abstract

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.

This is a preview of subscription content, access via your institution.

Fig. 1

Notes

  1. Or geometry; I shall not distinguish between these.

  2. Alternatively, if a connected submanifold M didn’t separate, one would construct a closed curve transverse to M lying in a neighborhood of M, and then argue that this would contradict the simple connectivity of the sphere (by homotopy invariance of intersection numbers).

  3. In some cases the connection between the homotopy theory and the geometric problem is sufficiently tight that any new geometric complexity statement would readily imply some, although perhaps not optimal, result about the complexity of homotopies.

  4. Many classical problems in analysis are also solved by homotopy methods. Typically, one starts, though, with an explicit homotopy to apply them to. I hope that the ideas discussed in later sections can be of use in that setting.

  5. Unless something extraordinary is happening—say f is locally constant.

  6. This can be achieved using either of the subdivision schemes of [23] or [30].

  7. The first level for \(E = \int \langle \nabla f, \nabla f \rangle / \int \langle f, f \rangle \) where i-dimensional mod 2 homology appears in the projective space of the Sobolev space \(H^{1,2}(M)\) is in the ith positive eigenvalue of the Laplacian.

  8. The reader could well want to know what can distinguish manifolds which mutually approach one another. The simplest invariants are odd primary characteristic classes of the topological tangent bundle. However, there are additional subtle secondary invariants that are torsion analogues of the invariants used by Atiyah and Bott [1] to distinguish lens spaces from one another.

  9. The process described, though, is useful even when \(B\Gamma \) does not exist as a finite complex. The homology described can be used as a substitute that occasionally has more useful properties. See [20] for an example of this.

  10. We shall not discuss here what to do about noise or when populations have overlapping images.

  11. Of course, in the noncompact case, Morse theory in its ordinary sense requires a properness condition, or a Palais–Smale condition in the infinite-dimensional setting.

  12. Other applications to approximation theory can be found in [49].

  13. Except the global maximum, which doesn’t close any \(H_0\) bar. Thought of as being the circle, it begins the nontrivial \(H_1\) bar.

  14. See the proof for a definition, or see, e.g., the Wikipedia page on Dehn functions of groups.

  15. Here I mean a large scale idea of texture that is analogous to the differences that one would notice on the small scale if one took the graphs of (Baire generic) Hölder functions for different exponents. If you consider the persistence homology, the Hölder exponent would be apparent in the “length spectrum of the bars” as will be explained in detail in a future paper with Yuliy Baryshnikov (and is not hard to see by wrinkling a map by putting in as many bumps at small scale as permitted by dimension and the Hölder condition).

  16. This linearity conjecture implies an estimate for \(\eta \)-invariants of manifolds (spectral invariants of odd-dimensional manifolds defined by Atiyah–Patodi–Singer) in terms of volume; Cheeger and Gromov proved these directly [13]. While the quantitative cobordism work we discuss is not strong enough to give the Cheeger–Gromov inequality, a different approach—with very interesting complexity applications—was given by [14].

  17. Actually, there is a beautiful proof of Thom’s theorem [6] that avoids this. This method actually directly leads to a polynomial estimate (of degree \(2^n\)) for k(W). Unlike what we are about to discuss, this does not extend to the oriented case.

  18. To see the issue, the number of point inverses for a Lipschitz map from \(S^1\) to itself can we be infinite, even with \(L = 2\). However, the degree of a Lipschitz map is bounded by L and one can \(C^0\)-approximate such a map by one where the number inverse images grows linearly with L.

  19. The construction we are about to explain is called the Whitehead product in homotopy theory. (See, e.g., [54].)

  20. However, for \({\text {Lip}}(S^1:M)\) for M a 3-dimensional Sol manifold the diameter is \(\exp (L)\), as suggested by a worst case analysis based on covering numbers.

References

  1. M. Atiyah and R. Bott. A Lefschetz fixed point formula for elliptic complexes: II. Applications. Ann. of Math., 88:451–491, 1968.

    MathSciNet  MATH  Article  Google Scholar 

  2. N. Amenta and M. Bern. Surface reconstruction by voronoi filtering. Discrete Comput. Geom., 22:481–504, 1999.

    MathSciNet  MATH  Article  Google Scholar 

  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. J. Boissonnat, F. Chazal, and M. Yvinec. Computational geometry and topology for data analysis. To appear.

  5. J. Boissonnat, L. Guibas, and S. Oudot. Manifold reconstruction in arbitrary dimensions using witness complexes. Discrete Comput. Geom., 42(1):37–70, 2009.

    MathSciNet  MATH  Article  Google Scholar 

  6. S. Buoncristiano and D. Hacon. An elementary geometric proof of two theorems of Thom. Topology, 20(1):97–99, 1981.

    MathSciNet  MATH  Article  Google Scholar 

  7. J. Bourgain. On Lipschitz embedding of finite metric spaces in Hilbert space. Israel J. Math., 52:46–52, 1985.

    MathSciNet  MATH  Article  Google Scholar 

  8. E. Brown. Finite computability of Postnikov complexes. Ann. of Math., 65(1):1–20, 1957.

    MathSciNet  MATH  Article  Google Scholar 

  9. W. Browder. Surgery on simply connected manifolds. Springer Verlag, 1972.

  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.

  11. G. Chambers, D. Dotterer, F. Manin, and S. Weinberger. Quantitative null-cobordism. J. Amer. Math. Soc., 31(4):1165–1203, 2018.

    MathSciNet  MATH  Article  Google Scholar 

  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.

  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.

    Chapter  Google Scholar 

  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.

    MathSciNet  MATH  Article  Google Scholar 

  15. G. Carlsson and F. Memoli. Classifying clustering schemes. Found. Comput. Math., 13:221–252, 2013.

    MathSciNet  MATH  Article  Google Scholar 

  16. G. Chambers, F. Manin, and S. Weinberger. Quantitative null homotopy and rational homotopy type. Geom. Funct. Anal., 28(3):563–588, 2018.

    MathSciNet  MATH  Article  Google Scholar 

  17. D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete Comput. Geom., 37(1):103–120, 2007.

    MathSciNet  MATH  Article  Google Scholar 

  18. G. Carlsson and A. Zomorodian. Computing persistent homology. Discrete Comput. Geom., 33(2):249–274, 2005.

    MathSciNet  MATH  Article  Google Scholar 

  19. F. Cucker and D. Zhou. Learning Theory: An Approximation Theory Viewpoint. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 2007.

  20. A. Dranishnikov, S. Ferry, and S. Weinberger. Large Riemannian manifolds which are flexible. Ann. of Math., 157:919–938, 2003.

    MathSciNet  MATH  Article  Google Scholar 

  21. A. Dranishnikov, S. Ferry, and S. Weinberger. An infinite dimensional phenomenon in finite dimensional topology. Preprint, 2017.

  22. M. DoCarmo. Riemannian Geometry. Birkhäuser Verlag, 1992.

  23. H. Edelsbrunner and D. Grayson. Edgewise subdivision of a simplex. Discrete Comput. Geom., 24(4):707–719, 2000.

    MathSciNet  MATH  Article  Google Scholar 

  24. H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete Comput. Geom., 28:511–533, 2002.

    MathSciNet  MATH  Article  Google Scholar 

  25. Y. Eliashberg and N. Mishachev. Introduction to the h-Principle, volume 48 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002.

  26. C. Fefferman. Whitney’s extension problem for \(c^m\). Ann. of Math., 164:313–359, 2006.

    MathSciNet  MATH  Article  Google Scholar 

  27. C. Fefferman. \(c^m\)-extension by linear operators. Ann. of Math., 166:779–835, 2007.

    MathSciNet  MATH  Article  Google Scholar 

  28. S. Ferry. Topological finiteness theorems for manifolds in Gromov-Hausdorff space. Duke Math. J., 74(1):95–106, 1994.

    MathSciNet  MATH  Article  Google Scholar 

  29. C. Fefferman, S. Mitter, and H. Naryanan. Testing the manifold hypothesis. J. Amer. Math. Soc., 29:983–1049, 2016.

    MathSciNet  MATH  Article  Google Scholar 

  30. S. Ferry and S. Weinberger. Quantitative algebraic topology and Lipschitz homotopy. Proc. Natl. Acad. Sci. U.S.A., 110:19246–19250, 2013.

    MathSciNet  MATH  Article  Google Scholar 

  31. M. Gromov and L. Guth. Generalizations of the Kolmogorov-Barzdin embedding estimates. Duke Math. J., 161:2549–2603, 2012.

    MathSciNet  MATH  Article  Google Scholar 

  32. M. Gromov. Homotopical effects of dilation. J. Diff. Geo., 13:313–310, 1978.

    Google Scholar 

  33. M. Gromov. Groups of polynomial growth and expanding maps. Publ. Math. IHÉS, 53:51–78, 1981.

    Article  Google Scholar 

  34. M. Gromov. Partial Differential Relations. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer Verlag, 1986.

  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. M. Gromov. Metric Structures for Riemannian and non-Riemannian Spaces. Modern Birkhäuser Classics. Birkhäuser Verlag, 1999.

  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.

  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.

  39. F. Manin. Plato’s cave and differential forms. Preprint, 2018.

  40. J. Matousek. Lecture notes on metric embeddings. Preprint, 2013.

  41. J. Milnor. Morse Theory, volume 51 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1963.

  42. J. Milnor. A note on curvature and fundamental group. J. Differ. Geom., 2:1–7, 1968.

    MathSciNet  MATH  Article  Google Scholar 

  43. F. Manin and S. Weinberger. The Gromov-Guth embedding theorem. Appendix to [11].

  44. A. Nabutovsky. Non-recursive functions, knots “with thick ropes” and self-clenching “thick” hyperspheres. Commun. Pure Appl. Math., 48(4):1–50, 1995.

    MathSciNet  MATH  Google Scholar 

  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.

  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.

    MathSciNet  MATH  Article  Google Scholar 

  47. A. Nabutovsky and S. Weinberger. Variational problems for Riemannian functionals and arithmetic groups. Publ. Math. IHÉS, 92:5–62, 2000.

    MathSciNet  MATH  Google Scholar 

  48. S. Oudot. Persistence Theory: From Quiver Representations to Data Analysis, volume 209 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2015.

  49. I. Polterovich, L. Polterovich, and V. Stojisavljević. Persistence barcodes and Laplace eigenfunctions on surfaces. Preprint, 2017.

  50. L. Polterovich, D. Rosen, K. Samvelyan, and J. Zhang. Persistent homology for symplectic topologists. Preprint, 2018.

  51. V. Robins. Toward computing homology from finite approximations. Topology Proceedings, 24:503–532, 1999.

    MathSciNet  MATH  Google Scholar 

  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.

  53. S. Smale. The classification of immersions of spheres in Euclidean spaces. Ann. of Math., 69:327–344, 1959.

    MathSciNet  MATH  Article  Google Scholar 

  54. E. Spanier. Algebraic Topology. McGraw-Hill, 1956.

  55. R. Strong. Notes on cobordism theory. Mathematical Notes. Princeton University Press, Princeton, NJ, 1968.

    Google Scholar 

  56. D. Sullivan. Infinitesimal computations in topology. Publ. Math. IHÉS, 47:269–331, 1977.

    MathSciNet  MATH  Article  Google Scholar 

  57. R. Thom. Quelques propriétés globales des variétés différentiables. Comment. Math. Helv., 28:17–86, 1954.

    MathSciNet  MATH  Article  Google Scholar 

  58. J. Traub and A. Werschulz. Complexity and information. Lezioni Lincee. Cambridge University Press, Cambridge, United Kingdom, 1999.

  59. L. Valiant. A theory of the learnable. Commun. ACM, 27(11):1134–1142, 1984.

    MATH  Article  Google Scholar 

  60. C. T. C. Wall. Surgery on Compact Manifolds. London Mathematical Society Monographs. Academic Press, Cambridge, MA, 1969.

  61. S. Weinberger. Computers, Rigidity, and Moduli. Princeton University Press, Princeton, NJ, 2004.

    Google Scholar 

  62. S. Weinberger. What is... persistent homology. Notices Amer. Math. Soc., 58(1), 2011.

  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.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Shmuel Weinberger.

Additional information

Communicated by Martin Sombra.

Publisher's Note

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

Shmuel Weinberger: Partially supported by an NSF Grant.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Weinberger, S. Interpolation, the Rudimentary Geometry of Spaces of Lipschitz Functions, and Geometric Complexity. Found Comput Math 19, 991–1011 (2019). https://doi.org/10.1007/s10208-019-09416-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-019-09416-0

Keywords

  • Interpolation
  • Function space
  • Persistent homology
  • Homotopy
  • Quantitative cobordism
  • Embedding

Mathematics Subject Classification

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