Topological data analysis is already established as a computer science discipline and applies results from the application of homological algebra to the filtration of a point set. The intersection between computer science, mathematics and statistics awakens many techniques that can be applied extremely broadly and across all scientific fields. Therefore, a short note that the book Geometric and Topological Inference by Jean-Daniel Boissonnat, Frédéric Chazal and Mariette Yvinec, provides a complete overview of the current state of the art and algorithmically represents persistent homology, with the required data structures. Furthermore, this book is an elegant introduction to computational topology. It is suitable for an entrance, especially for students with a computer science background, not trained in abstract mathematics.
Now that the NeuRIPS 2020 workshop Topological Data Analysis and Beyond has made clear the effort to tie the machine learning community closer to the topological data analysis community and expanded on the success of combining the two methods in a natural way (mathematically speaking), I'd like to share a few additional resources. Where does one start to learn persistent homology? How do I get an intuition for it? To begin with, Jänich's book on topology is an excellent choice for getting started and for the basic concepts. Enclosed I list equally excellent resources for learning algebraic topology, which is indispensable for a holistic understanding. These books are a very personal selection that I used to get into the subject.
- Klaus Jänich: Topologie. Springer.
- Allen Hatcher: Algebraic Topology.
- Herbert Edelsbrunner and John Harer: Computational Topology: An Introduction.
- David Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry.
- Scott Osbourne: Basic Homological Algebra. Springer.
- Steve Oudot: Persistence Theory: From Quiver Representation to Data Analysis.