We will use this page to share any kind of material that helped us to understand different topics in TDA. Please send me anything that you would like to share with the rest of the group and I will upload it here.

The Basics

  • a discussion on TDA references: [ link1 | link2 ]

  • Some references with Fabrizio’s favorite introductions to Persistent Homology
    1. Xiaojin Zhu, Persistent Homology: An Introduction and a New Text Representation for Natural Language Processing [ slides | pdf ]
    2. The supplement of Giseon Heo, Jennifer Gamble & Peter T. Kim, Topological Analysis of Variance and the Maxillary Complex [ supplement | link ]
    3. Our papers:
      • Confidence sets for persistence diagrams [ arXiv ]
      • Minimax Rates for Homology Inference [ arXiv ]

  • The notes of Fred’s course on Computational Geometry [ link ]

  • Robert Ghrist’s book, Elementary Applied Topology [ link ]

  • Herbert Edelsbrunner and John Harer, Computational Topology [ pdf ]

  • James R. Munkres, Elements Of Algebraic Topology, Ch 1.

  • Video lectures:
    1. Jessi recommends these video lectures from an introductory course in Algebraic Topology, given by N J Wildberger at UNSW, Sydney, Australia.
    2. Jisu found these video lectures on TDA by Isabel Darcy, University of Iowa.
    3. Gunnar Carlsson, The Shape of Data

Stability of Persistence Homology

  • F. Chazal, D. Cohen-Steiner, L. J. Guibas, F. Memoli, S. Y. Oudot, Gromov-Hausdorff Stable Signatures for Shapes using Persistence [ pdf ]
  • Herbert Edelsbrunner and John Harer, Computational Topology, p.216-222 [ pdf ]

Persistence Images

  • the Duke folks use “binned” diagrams: Paul Bendich, Sang Chin, Jesse Clarke, Jonathan deSena, John Harer, Elizabeth Munch, Andrew Newman, David Porter, David Rouse, Nate Strawn, Adam Watkins, Topological and Statistical Behavior Classifiers for Tracking Applications [ arXiv:1406.0214 ]
  • “vectorized persistence images” from folks at Colorado State + Lori Z: Henry Adams, Sofya Chepushtanova, Tegan Emerson, Eric Hanson, Michael Kirby, Francis Motta, Rachel Neville, Chris Peterson, Patrick Shipman, Lori Ziegelmeier, Persistence Images: A Stable Vector Representation of Persistent Homology [ slides | arXiv:1507.06217 ]

Topology of random complexes

  • Omer Bobrowski, Matthew Kahle, Topology of random complexes: a survey [ arXiv ]
  • Robert J. Adler, Omer Bobrowski, Matthew S. Borman, Eliran Subag, Shmuel Weinberger, Persistent Homology for Random Fields and Complexes [ arXiv ]
  • Omer Bobrowski, Robert J. Adler, Distance Functions, Critical Points, and the Topology of Random Cech Complexes [ arXiv ]
  • Omer Bobrowski, Sayan Mukherjee, The Topology of Probability Distributions on Manifolds [ arXiv ]
  • D. Yogeshwaran, Eliran Subag, Robert J. Adler, Random geometric complexes in the thermodynamic regime [ arXiv ]

Tree

  • Justin Eldridge, Mikhail Belkin, Yusu Wang, Beyond Hartigan Consistency: Merge Distortion Metric for Hierarchical Clustering [ arXiv:1506.06422 ]

Density based clustering

  • Bharath K. Sriperumbudur, Ingo Steinwart, Consistency and Rates for Clustering with DBSCAN [ pdf ]
  • Jeremy F. Magland, Alex H. Barnett, Unimodal clustering using isotonic regression: ISO-SPLIT [ arXiv:1508.04841 ]

Miscellaneous

  • Diego Hernan Diaz Martinez, Facundo Memoli, Washington Mio, The Shape of Data and Probability Measures [ arXiv:1509.04632 ]
  • Jose R. Berrendero, Antonio Cuevas, Beatriz Pateiro-Lopez, Shape classification based on interpoint distance distributions [ pdf ]

Applications in astronomy

  • Pratyush Pranav, Persistent Holes in the Universe: A hierarchical topology of the cosmic mass distribution(his thesis) [ pdf ]