2.2. Persistence Diagram Featurization

Persistence diagram can not be directly used in the machine learning algorithms. Therefore, there are several methods that are proposed to extract features from persistence diagrams. Some these methods are Persistence Images[1], Carlsson Coordinates[2, 6], Persistence Landscapes[3], Path Signatures[4, 5], template functions [8], and kernel method [9]. In this toolbox, we provide the documentation for the codes of these five methods. We used these methods to extract features from persistence diagrams of cutting signals to diagnose chatter in machining. Our data set is available in Ref. [7]. One can refer to [10] for more details.

• 2.2.1. Datasets
  • mnist()
  • mpeg7()
  • shrec14()
• 2.2.2. Featurization
  • 2.2.2.1. Persistence Landscapes
    • 2.2.2.1.1. Landscape class
      • PLandscape
    • 2.2.2.1.2. Feature matrix generation
      • F_Landscape()
  • 2.2.2.2. Persistence Images
    • F_Image()
    • 2.2.2.2.1. Computing Features from Persistence Images
  • 2.2.2.3. Carlsson Coordinates
    • F_CCoordinates()
  • 2.2.2.4. Template Functions
    • tent()
    • interp_polynomial()
  • 2.2.2.5. Path Signatures
    • F_PSignature()
  • 2.2.2.6. Kernel Method
    • KernelMethod()
• 2.2.3. Classification
  • getPercentScore()
  • 2.2.3.1. Persistence Landscapes
    • 2.2.3.1.1. Precomputed Landscapes
    • 2.2.3.1.2. Transfer learning between two sets of persistence diagrams
    • 2.2.3.1.3. Hyperparameter tuning
  • 2.2.3.2. Persistence Images
    • 2.2.3.2.1. Transfer learning between two sets of persistence diagrams
    • 2.2.3.2.2. Hyperparameter tuning
  • 2.2.3.3. Carlsson Coordinates
    • 2.2.3.3.1. Transfer learning between two sets of persistence diagrams
    • 2.2.3.3.2. Hyperparameter tuning
  • 2.2.3.4. Template Functions
    • 2.2.3.4.1. Transfer learning between two sets of persistence diagrams
    • 2.2.3.4.2. Hyperparameter tuning
  • 2.2.3.5. Path Signatures
  • 2.2.3.6. Kernel Method

2.2.4. References

[1]

Henry Adams, Tegan Emerson, Michael Kirby, Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. Persistence images: a stable vector representation of persistent homology. The Journal of Machine Learning Research, 18(1):218–252, 2017.

[2]

Aaron Adcock, Erik Carlsson, and Gunnar Carlsson. The ring of algebraic functions on persistence bar codes. Homology, Homotopy and Applications, 18(1):381–402, 2016. doi:10.4310/hha.2016.v18.n1.a21.

[3]

Peter Bubenik and Paweł Dłotko. A persistence landscapes toolbox for topological statistics. Journal of Symbolic Computation, 78:91–114, jan 2017. doi:10.1016/j.jsc.2016.03.009.

[4]

Ilya Chevyrev and Andrey Kormilitzin. A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788, 2016.

[5]

Ilya Chevyrev, Vidit Nanda, and Harald Oberhauser. Persistence paths and signature features in topological data analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(1):192–202, jan 2020. doi:10.1109/tpami.2018.2885516.

[6]

Firas A. Khasawneh, Elizabeth Munch, and Jose A. Perea. Chatter Classification in Turning using Machine Learning and Topological Data Analysis. IFAC-PapersOnLine, 51(14):195–200, 2018. doi:10.1016/j.ifacol.2018.07.222.

[7]

Firas A. Khasawneh, Andreas Otto, and Melih C. Yesilli. Turning dataset for chatter diagnosis using machine learning. 2019. doi:10.17632/HVM4WH3JZX.1.

[8]

Jose A. Perea, Elizabeth Munch, and Firas A. Khasawneh. Approximating continuous functions on persistence diagrams using template functions. arXiv preprint: 1902.07190, February 2019. arXiv:1902.07190.

[9]

Jan Reininghaus, Stefan Huber, Ulrich Bauer, and Roland Kwitt. A stable multi-scale kernel for topological machine learning. In 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, jun 2015. doi:10.1109/cvpr.2015.7299106.

[10]

Melih C. Yesilli, Firas A. Khasawneh, and Andreas Otto. Topological feature vectors for chatter detection in turning processes. arXiv preprint: 1905.08671, 2019.