Topology

From Ioannis Kourouklides
Jump to navigation Jump to search

This page contains resources about Geometric Topology and Topology in general, including Topological Data Analysis.

Subfields and Concepts

  • Metric Geometry 
    • Metric / Distance function
    • Geodesics
  • Topological Groups
    • Pontryagin duality (in Harmonic Analysis)
    • Locally Compact Abelian Group (LCA Group or LCAG)
  • Topological Spaces
    • Manifold
      • Riemannian Manifolds
    • Metric space
      • Riemannian metric
      • Fisher information metric / Fisher–Rao metric
  • Topological Data Analysis
    • barcode / persistence diagram
    • persistent homology / topological persistence

Online Courses

Video Lectures

Lecture Notes

Books

  • Boissonnat, J. D., Chazal, F., & Yvinec, M. (2018). Geometric and Topological Inference. Cambridge University Press. (link)
  • Tierny, J. (2018). Introduction to topological data analysis. UPMC, LIP6. (link)
  • Oudot, S. Y. (2015). Persistence Theory: From Quiver Representations to Data Analysis . American Mathematical Society.
  • Ghrist, R. W. (2014). Elementary applied topology. Createspace. (link)
  • Edelsbrunner, H., & Harer, J. (2010). Computational Topology: An Introduction. American Mathematical Society. (link)
  • Hatcher, A. (2002). Algebraic Topology. Cambridge University Press. (link)

Scholarly Articles

  • Carriere, M., Michel, B., & Oudot, S. (2018). Statistical analysis and parameter selection for Mapper. Journal of Machine Learning Research, 19(1), 478-516.
  • Wasserman, L. (2018). Topological data analysis. Annual Review of Statistics and Its Application, 5, 501-532.
  • Chazal, F., & Michel, B. (2017). An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists. arXiv preprint arXiv:1710.04019.
  • Chazal, F., Fasy, B., Lecci, F., Michel, B., Rinaldo, A., Rinaldo, A., & Wasserman, L. (2017). Robust topological inference: Distance to a measure and kernel distance. Journal of Machine Learning Research, 18(1), 5845-5884.
  • Hofer, C., Kwitt, R., Niethammer, M., & Uhl, A. (2017). Deep learning with topological signatures. In Advances in Neural Information Processing Systems (pp. 1634-1644).
  • Adams, H., Emerson, T., Kirby, M., Neville, R., Peterson, C., Shipman, P., ... & Ziegelmeier, L. (2017). Persistence images: A stable vector representation of persistent homology. Journal of Machine Learning Research, 18(1), 218-252.
  • Seversky, L. M., Davis, S., & Berger, M. (2016). On time-series topological data analysis: New data and opportunities. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (pp. 59-67).
  • Kusano, G., Hiraoka, Y., & Fukumizu, K. (2016). Persistence weighted Gaussian kernel for topological data analysis. In Proceedings of the 33rd International Conference on Machine Learning (pp. 2004-2013).
  • Chazal, F., Fasy, B. T., Lecci, F., Michel, B., Rinaldo, A., & Wasserman, L. (2015). Subsampling methods for persistent homology. In Proceedings of the 32nd International Conference on Machine Learning (pp. 2143-2151).
  • Chazal, F., Glisse, M., Labruere, C., & Michel, B. (2015). Convergence rates for persistence diagram estimation in topological data analysis. Journal of Machine Learning Research, 16(1), 3603-3635.
  • Bubenik, P. (2015). Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research, 16(1), 77-102.
  • Reininghaus, J., Huber, S., Bauer, U., & Kwitt, R. (2015). A stable multi-scale kernel for topological machine learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (pp. 4741-4748).
  • Kwitt, R., Huber, S., Niethammer, M., Lin, W., & Bauer, U. (2015). Statistical topological data analysis-a kernel perspective. In Advances in Neural Information Processing Systems (pp. 3070-3078).

Software

See also

Other Resources