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Persistence diagrams with linear machine learning models

  • Ippei ObayashiEmail author
  • Yasuaki Hiraoka
  • Masao Kimura
Article

Abstract

Persistence diagrams have been widely recognized as a compact descriptor for characterizing multiscale topological features in data. When many datasets are available, statistical features embedded in those persistence diagrams can be extracted by applying machine learnings. In particular, the ability for explicitly analyzing the inverse in the original data space from those statistical features of persistence diagrams is significantly important for practical applications. In this paper, we propose a unified method for the inverse analysis by combining linear machine learning models with persistence images. The method is applied to point clouds and cubical sets, showing the ability of the statistical inverse analysis and its advantages.

Keywords

Topological data analysis Persistent homology Machine learning Linear models Persistence image 

Mathematics Subject Classification

55-04 55U99 62P35 62J07 

Notes

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Supplementary material

References

  1. Adams, H., Chepushtanova, S., Emerson, T., Hanson, E., Kirby, M., Motta, F., Neville, R., Peterson, C., Shipman, P., Ziegelmeier, L.: Persistence images: a stable vector representation of persistent homology. J. Mach. Learn. Res. 18(8), 1–35 (2017)MathSciNetzbMATHGoogle Scholar
  2. Bauer, U., Kerber, M., Reininghaus, J.: Distributed computation of persistent homology. Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments (ALENEX) (2014)Google Scholar
  3. Bauer, U., Kerber, M., Reininghaus, J., Wagner, H.: Phat—persistent homology algorithms toolbox. J. Symb. Comput. 78, 76–90 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bingham, N.H., Fry, J.M.: Regression—Linear Models in Statistics. Springer, Berlin (2010)zbMATHGoogle Scholar
  5. Bishop, C.M.: Pattern Recognition and Machine Learning (Information Science and Statistics). Springer, Berlin (2007)Google Scholar
  6. Bubenik, P.: Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res. 16(1), 77–102 (2015)MathSciNetzbMATHGoogle Scholar
  7. Buchet, M., Hiraoka, Y., Obayashi, I.: Persistent homology and materials informatics. In: Tanaka, I. (ed.) Nanoinformatics, pp. 75–95. Springer, Berlin (2018)CrossRefGoogle Scholar
  8. Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46, 255–308 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Chazal, F., Glisse, M., Labruére, C., Michel, B.: Convergence rates for persistence diagram estimation in topological data analysis. J. Mach. Learn. Res. 16, 3603–3635 (2015)MathSciNetzbMATHGoogle Scholar
  10. Chan, J.M., Carlsson, G., Rabadan, R.: Topology of viral evolution. PNAS 110(46), 18566–18571 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discret. Comput. Geom. 37(1), 103–120 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Csurka, G., Bray, C., Dance, C. Fan, L.: Visual categorization with bags of keypoints. In: Proceeding of ECCV Workshop on Statistical Learning in Computer Vision, pp. 59–74 (2004)Google Scholar
  13. Da, T.K.F., Loriot, S., Yvinec, M.: 3D Alpha Shapes. CGAL User and Reference Manual 4.11, CGAL Editorial Board (2017)Google Scholar
  14. Delgado-Friedrichs, O., Robins, V., Sheppard, A.: Morse theory and persistent homology for topological analysis of 3D images of complex materials. In: 2014 IEEE International Conference on Image Processing (ICIP), pp. 4872–4876 (2014)Google Scholar
  15. Delgado-Friedrichs, O., Robins, V., Sheppard, A.: Skeletonization and partitioning of digital images using discrete Morse theory. IEEE Trans. Pattern Anal. Mach. Intell. 37(3), 654–666 (2015)CrossRefGoogle Scholar
  16. de Silva, V., Ghrist, R.: Coverage in sensor networks via persistent homology. Algebraic Geom. Topol. 7, 339–358 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Dey, T.K., Hirani, A.N., Krishnamoorthy, B.: Optimal homologous cycles, total unimodularity and linear programming. SIAM J. Comput. 40(4), 1026–1044 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discret. Comput. Geom. 28(4), 511–533 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. AMS, Providence (2010)zbMATHGoogle Scholar
  20. Escolar, E.G., Hiraoka, Y.: Optimal cycles for persistent homology via linear programming. Optimization in the Real World Toward Solving Real-World Optimization Problems, pp. 79–96. Springer Japan, Osaka (2016)Google Scholar
  21. Fasy, B.T., Lecci, F., Rinaldo, A., Wasserman, L., Balakrishnan, S., Singh, A.: Confidence sets for persistence diagrams. Ann. Stat. 42(6), 2301–2339 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Hiraoka, Y., Nakamura, T., Hirata, A., Escolar, E.G., Matsue, K., Nishiura, Y.: Hierarchical structures of amorphous solids characterized by persistent homology. Proc. Nat. Acad. Sci. USA 113, 7035–7040 (2016)CrossRefGoogle Scholar
  23. Ichinomiya, T., Obayashi, I., Hiraoka, Y.: Persistent homology analysis of craze formation. Phys. Rev. E 95(1), 012504 (2017)CrossRefGoogle Scholar
  24. Jones, E., Oliphant, T., Peterson, .P, et al.: SciPy: Open source scientific tools for Python. http://www.scipy.org/ (2001–) [Online; accessed 2018-01-20]
  25. Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  26. Kimura, M., Obayashi, I., Takeuchi, Y., Hiraoka, Y.: Finding trigger sites in heterogeneous reactions using persistent-homology without preliminary material scientific information. Sci. Rep. 8, 3553 (2018)CrossRefGoogle Scholar
  27. Kusano, G., Fukumizu, K., Hiraoka, Y.: Persistence weighted Gaussian kernel for topological data analysis. Proceedings of the 33rd International Conference on Machine Learning, JMLR: W&CP 48. 2004-2013 (2016)Google Scholar
  28. Kusano, G., Fukumizu, K., Hiraoka, Y.: Kernel method for persistence diagrams via kernel embedding and weight factor. Accepted in Journal of Machine Learning ResearchGoogle Scholar
  29. Lowe, D.G.: Object recognition from local scale invariant features. In: Proc. of IEEE International Conference on Computer Vision, pp. 1150–1157 (1999)Google Scholar
  30. Nowak, E., Jurie, F., Triggs, B.: Sampling Strategies for Bag-of-Features Image Classification. In: Computer Vision – ECCV 2006: 9th European Conference on Computer Vision, Graz, Austria, May 7-13, 2006, Proceedings, Part IV, pp. 490–503 (2006)Google Scholar
  31. Otter, N., Porter, M.A., Tillmann, U., Grindrod, P., Harrington, H.A.: A roadmap for the computation of persistent homology. arXiv:1506.08903
  32. Pearson, D.A., Bradley, R.M., Motta, F.C., Shipman, P.D.: Producing nanodot arrays with improved hexagonal order by patterning surfaces before ion sputtering. Phys. Rev. E 92(6), 062401 (2015)CrossRefGoogle Scholar
  33. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Erplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., Duchesnay, E.: Scikit-learn: Machine learning in Python. J. Mach. Learn. Res. 12, 2825–2830 (2011)MathSciNetzbMATHGoogle Scholar
  34. Rajan, K.: Materials informatics. Mater. Today 8(10), 38–45 (2005)CrossRefGoogle Scholar
  35. Rajan, K.: Materials informatics. Mater. Today 15(11), 470 (2012)CrossRefGoogle Scholar
  36. Reininghaus, J., Huber, S., Bauer, U., Kwitt, R.: A Stable Multi-Scale Kernel for Topological Machine Learning. 2015 IEEE Conference on Computer Vision and Pattern Recognition, 4741–4748 (2015)Google Scholar
  37. Robert, T.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B (Methodol.) 58(1), 267–288 (1996)MathSciNetzbMATHGoogle Scholar
  38. Robins, V., Turner, K.: Principal component analysis of persistent homology rank functions with case studies of spatial point patterns, sphere packing and colloids. Phys. D 334, 99–117 (2016)MathSciNetCrossRefGoogle Scholar
  39. Robins, V., Saadatfar, M., Delgado-Friedrichs, O., Sheppard, A.P.: Percolating length scales from topological persistence analysis of micro-CT images of porous materials. Water Resour. Res. 52(1), 315–329 (2016)CrossRefGoogle Scholar
  40. Saadatfar, M., Takeuchi, H., Francois, N., Robins, V., Hiraoka, Y.: Pore configuration landscape of granular crystallisation. Nat. Commun. 8, 15082 (2017).  https://doi.org/10.1038/ncomms15082 CrossRefGoogle Scholar
  41. Sivic, J. and Zisserman, A.: Video Google: A Text Retrieval Approach to Object Matching in Videos. In: Proc. of IEEE International Conference on Computer Vision, pp.1470–1477 (2003)Google Scholar
  42. Turner, K., Mileyko, Y., Mukherjee, S., Harer, J.: Fréchet means for distributions of persistence diagrams. Discret. Comput. Geom. 52(1), 44–70 (2014)CrossRefzbMATHGoogle Scholar
  43. Zomorodian, A., Carlsson, G.: Computing persistent homology. Discret. Comput. Geom. 33(2), 249–274 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Advanced Institute for Materials Research (WPI-AIMR)Tohoku UniversitySendaiJapan
  2. 2.Kyoto University Institute for Advanced StudyKyoto UniversityKyotoJapan
  3. 3.Center for Advanced Intelligence ProjectRIKENWakoJapan
  4. 4.Center for Materials research by Information Integration (CMI2)National Institute for Materials Science (NIMS)TsukubaJapan
  5. 5.Photon Factory, Institute of Materials Structure ScienceHigh Energy Accelerator Research OrganizationTsukubaJapan
  6. 6.Department of Materials Structure Science, School of High Energy Accelerator ScienceSOKENDAI (The Graduate University for Advanced Studies)TsukubaJapan

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