Topological Data Analysis: Developments and Applications

Topological Data Analysis (TDA) and its mainstay computational device, persistent homology (PH), has established a strong track record of providing researchers across the data-driven sciences with new insights and methodologies by characterizing low-dimensional geometric structures in high-dimensional data. When combined with machine learning (ML) methods, PH is valued as a discriminating-feature extraction tool. This work highlights many of the recent successes at the intersection of TDA and ML, introduces some of the foundational mathematics underpinning TDA, and summarizes the efforts to strengthen the bridge between TDA and ML. Thus, this document is a launching point for experimentalists and theoreticians to consider what can be learned from the shape of their data.

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Persistent Homology Techniques for Big Data and Machine Intelligence: A Survey

Persistent Homology in Data Science

Topology, Big Data and Optimization

Notes

The sublevel sets of a real-valued function, \(f: X \rightarrow \mathbb\) , are the subsets f −1 ((−∞, a]) = x ∈ X | f(x) ≤ a> ⊂ X.

The level sets of a real-valued function, \(f: X \rightarrow \mathbb\) , are the subsets f −1 (a>) = x ∈ X | f(x) = a> ⊂ X.

A set of vectors \(\ \subset \mathbb^\) is affinely independent if the set v _i − v ₀ | i = 1, …, n> is linearly independent. Often called the 1-skeleton. The trivial vector space over \(\mathbb\) consisting only of the 0 vector.

More generally, the sets of n-chains may be defined to be the free abelian groups with coefficients taken from a commutative ring. In this setting the boundary maps are homomorphisms (Hatcher, 2002).

If \(\mathbb\) is chosen to be a commutative ring, the boundaries and cycles form subgroups which explains the terminology homology groups.

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Authors and Affiliations

1. Mathematics Department, Duke University, Durham, NC, USA Francis C. Motta

1. Francis C. Motta