Heuristic Framework for Multiscale Testing of the Multi-Manifold Hypothesis

  • F. Patricia Medina
  • Linda NessEmail author
  • Melanie Weber
  • Karamatou Yacoubou Djima
Part of the Association for Women in Mathematics Series book series (AWMS, volume 17)


When analyzing empirical data, we often find that global linear models overestimate the number of parameters required. In such cases, we may ask whether the data lies on or near a manifold or a set of manifolds, referred to as multi-manifold, of lower dimension than the ambient space. This question can be phrased as a (multi-)manifold hypothesis. The identification of such intrinsic multiscale features is a cornerstone of data analysis and representation, and has given rise to a large body of work on manifold learning. In this work, we review key results on multiscale data analysis and intrinsic dimension followed by the introduction of a heuristic, multiscale, framework for testing the multi-manifold hypothesis. Our method implements a hypothesis test on a set of spline-interpolated manifolds constructed from variance-based intrinsic dimensions. The workflow is suitable for empirical data analysis as we demonstrate on two use cases.



This research started at the Women in Data Science and Mathematics Research Collaboration Workshop (WiSDM), July 17–21, 2017, at the Institute for Computational and Experimental Research in Mathematics (ICERM). The workshop was partially supported by grant number NSF-HRD 1500481-AWM ADVANCE and co-sponsored by Brown’s Data Science Initiative.

Additional support for some participant travel was provided by DIMACS in association with and through its Special Focus on Information Sharing and Dynamic Data Analysis. Linda Ness worked on this project during a visit to DIMACS, partially supported by the National Science Foundation under grant number CCF-1445755. F. Patricia Medina received partial travel funding from the Mathematical Science Department at Worcester Polytechnic Institute.

We thank Brie Finegold and Katherine M. Kinnaird for their participation in the workshop and in early stage experiments. In addition, we thank Anna Little for helpful discussions on intrinsic dimensions and Jason Stoker for sharing material on LiDAR data.


  1. 1.
    E. Arias-Castro, G. Chen, G. Lerman, Spectral clustering based on local linear approximations. Electr. J. Stat. 5, 1537–1587 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Azzam, R. Schul, An analyst’s traveling salesman theorem for sets of dimension larger than one. Tech Report (2017).
  3. 3.
    D. Bassu, R. Izmailov, A. McIntosh, L. Ness, D. Shallcross, Centralized multi-scale singular vector decomposition for feature construction in LiDAR image classification problems, in IEEE Applied Imagery and Pattern Recognition Workshop (AIPR) (IEEE, Piscataway, 2012)Google Scholar
  4. 4.
    D. Bassu, R. Izmailov, A. McIntosh, L. Ness, D. Shallcross, Application of multi-scale singular vector decomposition to vessel classification in overhead satellite imagery, in Proceedings of the Seventh Annual International Conference on Digital Image Processing (ICDIP 2015), vol. 9631, ed. by C. Falco, X. Jiang (2015)Google Scholar
  5. 5.
    M. Belkin, P. Niyogi, Laplacian Eigenmaps and spectral techniques for embedding and clustering, in Advances in Neural Information Processing Systems (NIPS), vol. 14 (2002)Google Scholar
  6. 6.
    M. Belkin, P. Niyogi, Laplacian Eigenmaps for dimensionality reduction and data representation. Neural Comput. 15, 1373–1396 (2002)CrossRefGoogle Scholar
  7. 7.
    M. Belkin, P. Niyogi, Laplacian Eigenmaps for dimensionality reduction and data representation. Neural Comput. 15, 1373–1396 (2003)CrossRefGoogle Scholar
  8. 8.
    P. Bendich, E. Gasparovic, J. Harer, R. Izmailov, L. Ness, Multi-scale local shape analysis and feature selection in machine learning applications, in Multi-Scale Local Shape Analysis and Feature Selection in Machine Learning Applications (IEEE, Piscataway, 2014). Google Scholar
  9. 9.
    P. Bendich, E. Gasparovic, C. Tralie, J. Harer, Scaffoldings and spines: organizing high-dimensional data using cover trees, local principal component analysis, and persistent homology. Technical Report (2016).
  10. 10.
    A. Beygelzimer, S. Kakade, J. Langford, Cover trees for nearest neighbor, in Proceedings of the 23rd International Conference on Machine Learning (ICML ’06) (ACM, New York 2006), pp. 97–104Google Scholar
  11. 11.
    N. Brodu, D. Lague, 3D terrestrial LiDAR data classification of complex natural scenes using a multi-scale dimensionality criterion: applications in geomorphology. ISPRS J. Photogramm. Remote Sens. 68, 121–134 (2012)CrossRefGoogle Scholar
  12. 12.
    F. Camastra, Data dimensionality estimation methods: a survey. Pattern Recognit. 36, 2945–2954 (2003)CrossRefGoogle Scholar
  13. 13.
    F. Camastra, A. Vinciarelli, Estimating the intrinsic dimension of data with a fractal-based method. IEEE Trans. Pattern Anal. Mach. Intell. 24, 1404–1407 (2002)CrossRefGoogle Scholar
  14. 14.
    K. Carter, A. Hero, Variance reduction with neighborhood smoothing for local intrinsic dimension estimation, in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, Piscataway, 2008)Google Scholar
  15. 15.
    G. Chen, A. Little, M. Maggioni, Multi-resolution geometric analysis for data in high dimensions, in Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center (Springer, Berlin, 2013), pp. 259–285Google Scholar
  16. 16.
    J. Chodera, W. Swope, J. Pitera, K. Dill, Long-time protein folding dynamics from short-time molecular dynamics simulations. Multiscale Model. Simul. 5, 1214–1226 (2006)MathSciNetCrossRefGoogle Scholar
  17. 17.
    R. Coifman, S. Lafon, Diffusion maps. Appl. Comput. Harmon. Anal. 21, 5–30 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    R. Coifman, S. Lafon, M. Maggioni, B. Nadler, F. Warner, S.W. Zucker, Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. Proc. Natl. Acad. Sci. U. S. A. 102, 7426–31 (2005)CrossRefGoogle Scholar
  19. 19.
    R.R. Coifman, I. Kevrekidis, S. Lafon, M. Maggioni, B. Nadler, Diffusion maps, reduction coordinates, and low dimensional representation of stochastic systems. Multiscale Model. Simul. 7, 842–864 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    J.A. Costa, A.O. Hero, Geodesic entropic graphs for dimension and entropy estimation in manifold learning. IEEE Trans. Signal Process. 52, 2210–2211 (2004)MathSciNetCrossRefGoogle Scholar
  21. 21.
    J.A. Costa, A. Girotra, A.O. Hero, Estimating local intrinsic dimension with k-nearest neighbor graphs, in IEEE/SP 13th Workshop on Statistical Signal Processing (IEEE, Piscataway, 2005)Google Scholar
  22. 22.
    G. David, S. Semmes, Quantitative rectifiability and Lipschitz mappings. Trans. Am. Math. Soc. 2, 855–889 (1993) MathSciNetCrossRefGoogle Scholar
  23. 23.
    D. Donoho, C. Grimes, Hessian eigenmaps: locally linear embedding techniques for high-dimensional data. Proc. Natl. Acad. Sci. U. S. A. 100, 5591–5596 (2003)MathSciNetCrossRefGoogle Scholar
  24. 24.
    C. Fefferman, S. Mitter, H. Narayanan, Testing the manifold hypothesis. J. Am. Math. Soc. 29, 983–1049 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    K. Fukunaga, Intrinsic dimensionality extraction, in Classification Pattern Recognition and Reduction of Dimensionality. Handbook of Statistics, vol. 2 (Elsevier, Amsterdam, 1982), pp. 347–360Google Scholar
  26. 26.
    P. Grassberger, I. Procaccia, Measuring the strangeness of strange attractors. Phys. D 9, 189–208 (1983)MathSciNetCrossRefGoogle Scholar
  27. 27.
    J. Ham, D. Lee, S. Mika, B. Schölkopf, A kernel view of the dimensionality reduction of manifolds, in Proceedings of the Twenty-First International Conference on Machine Learning (ICML ’04) (ACM, New York, 2004), pp. 47–55Google Scholar
  28. 28.
    G. Haro, G. Randall, G. Sapiro, Translated Poisson mixture model for stratification learning. Int. J. Comput. Vis. 80, 358–374 (2008)CrossRefGoogle Scholar
  29. 29.
    D. Joncas, M. Meila, J. McQueen, Improved graph Laplacian via geometric self-consistency, in Advances in Neural Information Processing Systems (2017), pp. 4457–4466Google Scholar
  30. 30.
    P.W. Jones, Rectifiable sets and the traveling salesman problem. Invent. Math. 102, 1–15 (1990)MathSciNetCrossRefGoogle Scholar
  31. 31.
    D.R. Karger, M. Ruhl, Finding nearest neighbors in growth-restricted metrics, in Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing (STOC ’02) (ACM, New York, 2002), pp. 741–750zbMATHGoogle Scholar
  32. 32.
    R. Krauthgamer, J.R. Lee, Navigating nets: simple algorithms for proximity search, in Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA ’04) (Philadelphia, Society for Industrial and Applied Mathematics, 2004), pp. 798–807Google Scholar
  33. 33.
    J. Lee, M. Verleysen, Nonlinear Dimensionality Reduction, 1st edn. (Springer, Berlin, 2007)CrossRefGoogle Scholar
  34. 34.
    E. Levina, P. Bickel, Maximum likelihood estimation of intrinsic dimension, in Advances in Neural Information Processing Systems (NIPS), vol. 17 (MIT Press, Cambridge, MA, 2005), pp. 777–784Google Scholar
  35. 35.
    A. Little, Estimating the Intrinsic Dimension of High-Dimensional Data Sets: A Multiscale, Geometric Approach, vol. 5 (Duke University, Durham, 2011)Google Scholar
  36. 36.
    P.M. Mather, Computer Processing of Remotely-Sensed Images: An Introduction (Wiley, New York, 2004)Google Scholar
  37. 37.
    J. McQueen, M. Meila, J. VanderPlas, Z. Zhang, Megaman: scalable manifold learning in python. J. Mach. Learn. Res. 17, 1–5 (2016)MathSciNetzbMATHGoogle Scholar
  38. 38.
    B. Nadler, S. Lafon, R. Coifman, I. Kevrekidis, Diffusion maps, spectral clustering and eigenfunctions of Fokker-Planck operators. Appl. Comput. Harmon. Anal. 21, 113–127 (2006)MathSciNetCrossRefGoogle Scholar
  39. 39.
    H. Narayanan, S. Mitter, Sample complexity of testing the manifold hypothesis, in Advances in Neural Information Processing Systems, vol. 23. ed. by J.D. Lafferty, C.K.I. Williams, J. Shawe-Taylor, R.S. Zemel, A. Culotta (Curran Associates, Red Hook, 2010), pp. 1786–1794Google Scholar
  40. 40.
    A. Ng, M. Jordan, Y. Weiss, On spectral clustering: analysis and an algorithm, in Advances in Neural Information Processing Systems (NIPS), vol. 14 (2002), pp. 849–856Google Scholar
  41. 41.
    K.W. Pettis, T.A. Bailey, A.K. Jain, R.C. Dubes, An intrinsic dimensionality estimator from near-neighbor information. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-1, 25–37 (1979)CrossRefGoogle Scholar
  42. 42.
    S.T. Roweis, L.K. Saul, Nonlinear dimensionality reduction by locally linear embedding. Science 290, 2323–2326 (2000)CrossRefGoogle Scholar
  43. 43.
    L.K. Saul, S.T. Roweis, Think globally, fit locally: unsupervised learning of low dimensional manifolds. J. Mach. Learn. Res. 4, 119–155 (2003)MathSciNetzbMATHGoogle Scholar
  44. 44.
    B. Schölkopf, A. Smola, J. Alexander, K. Müller, Kernel principal component analysis, in Advances in Kernel Methods: Support Vector Learning (1999), pp. 327–352Google Scholar
  45. 45.
    J. Shan, C.K. Toth, Topographic Laser Ranging and Scanning: Principles and Processing, 1st edn. (CRC Press, Boca Raton, 2008)CrossRefGoogle Scholar
  46. 46.
  47. 47.
    G. Sumerling, Lidar Analysis in Arcgis 9.3.1 for Forestry Applications. (2010)
  48. 48.
    F. Takens, On the Numerical Determination of the Dimension of an Attractor (Springer, Berlin, 1985), pp. 99–106zbMATHGoogle Scholar
  49. 49.
    J.B. Tenenbaum, V. de Silva, J.C. Langford, A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319–2323 (2000)CrossRefGoogle Scholar
  50. 50.
    J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, J.D. Farmer, Testing for nonlinearity in time series: the method of surrogate data. Phys. D: Nonlinear Phenom. 58, 77–94 (1992)CrossRefGoogle Scholar
  51. 51.
    J. Wang, A.L. Ferguson, Nonlinear reconstruction of single-molecule free-energy surfaces from univariate time series. Phys. Rev. E 93, 032412 (2016)CrossRefGoogle Scholar
  52. 52.
    X. Wang, K. Slavakis, G. Lerman, Riemannian multi-manifold modeling. Technical Report (2014). http://arXiv:1410.0095 and Link to supplementary webpage with code
  53. 53.
    W. Zheng, M. Rohrdanz, M. Maggioni, C. Clementi, Determination of reaction coordinates via locally scaled diffusion map. J. Chem. Phys. 134, 03B624 (2011)Google Scholar
  54. 54.
    W. Zjeng, M. Rohrdanz, C. Clementi, Rapid exploration of configuration space with diffusion-map-directed molecular dynamics. J. Phys. Chem. B 117, 12769–12776 (2013)CrossRefGoogle Scholar

Copyright information

© The Author(s) and the Association for Women in Mathematics 2019

Authors and Affiliations

  • F. Patricia Medina
    • 1
  • Linda Ness
    • 2
    Email author
  • Melanie Weber
    • 3
  • Karamatou Yacoubou Djima
    • 4
  1. 1.Worcester Polytechnic InstituteWorcesterUSA
  2. 2.Rutgers UniversityNew BrunswickUSA
  3. 3.Princeton UniversityPrincetonUSA
  4. 4.Amherst CollegeAmherstUSA

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