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Embeddings of Riemannian manifolds with finite eigenvector fields of connection Laplacian

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Abstract

We study the problem asking if one can embed manifolds into finite dimensional Euclidean spaces by taking finite number of eigenvector fields of the connection Laplacian. This problem is essential for the dimension reduction problem in manifold learning. In this paper, we provide a positive answer to the problem. Specifically, we use eigenvector fields to construct local coordinate charts with low distortion, and show that the distortion constants depend only on geometric properties of manifolds with metrics in the little Hölder space \(c^{2,\alpha }\). Next, we use the coordinate charts to embed the entire manifold into a finite dimensional Euclidean space. The proof of the results relies on solving the elliptic system and providing estimates for eigenvector fields and the heat kernel and their gradients. We also provide approximation results for eigenvector field under the \(c^{2,\alpha }\) perturbation.

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References

  1. Alexeev, B., Bandeira, A.S., Fickus, M., Mixon, D.G.: Phase retrieval with polarization. SIAM J. Imaging Sci. 7(1), 35–66 (2014). https://doi.org/10.1137/12089939X

    Article  MATH  Google Scholar 

  2. Anderson, M.: Convergence and rigidity of manifolds under Ricci curvature bounds. Invent. Math. 102, 429–445 (1990)

    Article  MathSciNet  Google Scholar 

  3. Anderson, M.T., Cheeger, J.: \(c^\alpha \)-compactness for manifolds with Ricci curvature and injectivity radius bounded below. J. Differ. Geom. 35, 265–281 (1992)

    Article  MathSciNet  Google Scholar 

  4. Bates, J.: The embedding dimension of Laplacian eigenfunction maps. Appl. Comput. Harmon. Anal. 37(3), 516–530 (2014)

    Article  MathSciNet  Google Scholar 

  5. Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15(6), 1373–1396 (2003)

    Article  Google Scholar 

  6. Bérard, P.: Spectral Geometry: Direct and Inverse Problems. Springer, New York (1986)

    Book  Google Scholar 

  7. Bérard, P., Besson, G., Gallot, S.: Embedding Riemannian manifolds by their heat kernel. Geom. Funct. Anal. 4, 373–398 (1994). https://doi.org/10.1007/BF01896401

    Article  MathSciNet  MATH  Google Scholar 

  8. Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Grundlehren Text Editions. Springer, Berlin (2004)

    MATH  Google Scholar 

  9. Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow. American Mathematical Society, New York (2006)

    Book  Google Scholar 

  10. Chung, F.: Spectral Graph Theory. American Mathematical Society, New York (1996)

    Book  Google Scholar 

  11. Chung, F., Kempton, M.: A local clustering algorithm for connection graphs. In: Bonato, A., Mitzenmacher, M., Pralat, P. (eds.) Algorithms and Models for the Web Graph, Volume 8305 of Lecture Notes in Computer Science, pp. 26–43. Springer, New York (2013)

    Google Scholar 

  12. Chung, F., Zhao, W., Kempton, M.: Ranking and sparsifying a connection graph. In: Bonato, A., Janssen, J. (eds.) Algorithms and Models for the Web Graph. Lecture Notes in Computer Science, vol. 7323, pp. 66–77. Springer, Berlin (2012)

    Chapter  Google Scholar 

  13. Coifman, R.R., Lafon, S.: Diffusion maps. Appl. Comput. Harmon. Anal. 21(1), 5–30 (2006)

    Article  MathSciNet  Google Scholar 

  14. Coulhon, T., Devyver, B., Sikora, A.: Gaussian heat kernel estimates: from functions to forms (2016). arXiv:1606.02423 [math.AP]

  15. Croke, C.: Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. Ecole Norm. Sup. 13, 419–435 (1980)

    Article  MathSciNet  Google Scholar 

  16. DeTurck, D., Kazdan, J.: Some regularity theorems in riemannian geometry. Ann. Sci. Ec. Norm. Sup. 14, 249–260 (1981)

    Article  MathSciNet  Google Scholar 

  17. Devyver, B.: A Gaussian estimate for the heat kernel on differential forms and application to the Riesz transform. Math. Ann. 358(1–2), 25–68 (2014)

    Article  MathSciNet  Google Scholar 

  18. Dolzmann, G., Múller, S.: Estimates for green’s matrices of elliptic systems by \(l^p\) theory. Manuscripta Mathematica 88(1), 261–273 (1995)

    Article  MathSciNet  Google Scholar 

  19. Donoho, D.L., Grimes, C.: Hessian eigenmaps: locally linear embedding techniques for high-dimensional data. Proc. Natl. Acad. Sci. USA 100(10), 5591–5596 (2003)

    Article  MathSciNet  Google Scholar 

  20. El Karoui, N., Wu, H.-T.: Connection graph Laplacian and random matrices with random blocks. Inf. Inference J. IMA 4, 1–42 (2015)

    Article  Google Scholar 

  21. El Karoui, N., Wu, H.-T.: Connection graph Laplacian methods can be made robust to noise. Ann. Stat. 44(1), 346–372 (2016)

    Article  Google Scholar 

  22. Gilkey, P.: The Index Theorem and the Heat Equation. Publish or Perish, inc., Boston (1974)

    MATH  Google Scholar 

  23. Hadani, R., Singer, A.: Representation theoretic patterns in three dimensional cryo-electron microscopy I: the intrinsic reconstitution algorithm. Ann. Math. 174(2), 1219–1241 (2011)

    Article  MathSciNet  Google Scholar 

  24. Hess, H., Schrader, R., Uhlenbrock, D.A.: Kato’s inequality and the spectral distribution of Laplacians on compact Riemannian Manifolfs. J. Differ. Geom. 15(1), 27–37 (1980)

    Article  Google Scholar 

  25. Jones, P., Maggioni, M., Schul, R.: Universal local parametrization via heat kernels and eigenfunctions of the Laplacian. Ann. Acad. Sci. Fenn. Math. 35, 131–174 (2010)

    Article  MathSciNet  Google Scholar 

  26. Kufner, A., John, O., Fucik, S.: Function Spaces. Noordhoff International Publishing, Sussex (1977)

    MATH  Google Scholar 

  27. Lederman, R.R., Talmon, R.: Learning the geometry of common latent variables using alternating-diffusion. Appl. Comput. Harmon. Anal. 44(3), 509–536. https://doi.org/10.1016/j.acha.2015.09.002

    Article  MathSciNet  Google Scholar 

  28. Li, P., Yau, S.T.: On the parabolic kernel of the schrödinger operator. Acta Math. 156(1), 153–201 (1986)

    Article  MathSciNet  Google Scholar 

  29. Marchesini, S., Tu, Y.-C., Wu, H.-T.: Alternating projection, ptychographic imaging and phase synchronization. Appl. Comput. Harmon. Anal. 41(3), 815–851 (2016)

    Article  MathSciNet  Google Scholar 

  30. Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer, New York (1966)

    MATH  Google Scholar 

  31. Nash, J.: The imbedding problem for Riemannian Manifolds. Ann. Math. 63(1), 20–63 (1956)

    Article  MathSciNet  Google Scholar 

  32. Palais, R.: Foundations of Global Non-Linear Analysis. Benjamin, New York (1968)

    MATH  Google Scholar 

  33. Portegies, J.W.: Embeddings of Riemannian manifolds with heat kernels and eigenfunctions. Commun. Pure Appl. Math. (2015). arXiv:1311.7568 [math.DG]

  34. Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500), 2323–2326 (2000)

    Article  Google Scholar 

  35. Singer, A., Coifman, R.R.: Non-linear independent component analysis with diffusion maps. Appl. Comput. Harmon. Anal. 25(2), 226–239 (2008)

    Article  MathSciNet  Google Scholar 

  36. Singer, A., Wu, H.-T.: Orientability and diffusion map. Appl. Comput. Harmon. Anal. 31(1), 44–58 (2011)

    Article  MathSciNet  Google Scholar 

  37. Singer, A., Wu, H.-T.: Vector diffusion maps and the connection Laplacian. Commun. Pure Appl. Math. 65(8), 1067–1144 (2012)

    Article  MathSciNet  Google Scholar 

  38. Singer, A., Wu, H.-T.: 2-d tomography from noisy projections taken at unknown random directions. SIAM J. Imaging Sci. 6(1), 136–175 (2013)

    Article  MathSciNet  Google Scholar 

  39. Singer, A., Wu, H.-T.: Spectral convergence of the connection Laplacian from random samples. Inf. Inference J. IMA 6(1), 58–123 (2017)

    MathSciNet  MATH  Google Scholar 

  40. Singer, A., Zhao, Z., Shkolnisky, Y., Hadani, R.: Viewing angle classification of cryo-electron microscopy images using eigenvectors. SIAM J. Imaging Sci. 4(2), 723–759 (2011)

    Article  MathSciNet  Google Scholar 

  41. Talmon, R., Coifman, R.R.: Empirical intrinsic geometry for intrinsic modeling and nonlinear filtering. Proc. Nat. Acad. Sci. 110(31), 12535–12540 (2013)

    Article  Google Scholar 

  42. Talmon, R., Wu, H.-T.: Latent common manifold learning with alternating diffusion: Analysis and applications. Appl. Comput. Harmon. Anal. (2018). https://doi.org/10.1016/j.acha.2017.12.006

  43. Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)

    Article  Google Scholar 

  44. Wang, C.: The calderón–zygmund inequality on a compact Riemannian manifold. Pac. J. Math. 217(1), 181–200 (2004)

    Article  MathSciNet  Google Scholar 

  45. Wang, X., Zhu, K.: Isometric embeddings via heat kernel. J. Differ. Geom. 99, 497–538 (2015)

    Article  MathSciNet  Google Scholar 

  46. Whitney, H.: The singularities of a smooth n-manifold in \((2n-1)\)-space. Ann. Math. 45, 247–293 (1944)

    Article  MathSciNet  Google Scholar 

  47. Wu, H.-T.: Embedding Riemannian Manifolds by the heat kernel of the connection Laplacian. Adv. Math. 304(2), 1055–1079 (2017)

    Article  MathSciNet  Google Scholar 

  48. Wu, H.-T., Wu, N.: Think globally, fit locally under the Manifold Setup: asymptotic analysis of locally linear embedding. Ann. Stat. (2018) (in press)

  49. Zhao, Z., Singer, A.: Rotationally invariant image representation for viewing direction classification in cryo-EM. J. Struct. Biol. 186(1), 153–166 (2014)

    Article  Google Scholar 

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Acknowledgements

Hau-tieng Wu’s research is partially supported by Sloan Research Fellow FR-2015-65363. Chen-Yun Lin would like to thank Thomas Nyberg for his helpful discussions. The authors acknowledge anonymous reviewers’ valuable comments.

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Authors and Affiliations

  1. Department of Mathematics, Duke University, Durham, USA

    Chen-Yun Lin

  2. Department of Mathematics and Department of Statistical Science, Duke University, Durham, USA

    Hau-Tieng Wu

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  1. Chen-Yun Lin
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Correspondence to Hau-Tieng Wu.

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Communicated by A. Chang.

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Lin, CY., Wu, HT. Embeddings of Riemannian manifolds with finite eigenvector fields of connection Laplacian. Calc. Var. 57, 126 (2018). https://doi.org/10.1007/s00526-018-1401-3

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