Persistent Homology and the Upper Box Dimension

Singular persistent homology with geometrically parallelizable computation

Persistent homology is a popular tool in Topological Data Analysis. It provides numerical characteristics of data sets which reflect global geometric properties. In order to be useful in practice, for example for feature generation in machine learning, it needs to be effectively computable. Classical homology is a computable topological invariant because of the Mayer-Vietoris exact and spectral sequences associated to coverings of a space. We state and prove versions of the Mayer-Vietoris theorem for persistent homology under mild and commonplace assumptions. This is done through the use of a …

The persistent homology of cyclic graphs

We give an $O(n^2(k+\log n))$ algorithm for computing the $k$-dimensional persistent homology of a filtration of clique complexes of cyclic graphs on $n$ vertices. This is nearly quadratic in the number of vertices $n$, and therefore a large improvement upon the traditional persistent homology algorithm, which is cubic in the number of simplices of dimension at most $k+1$, and hence of running time $O(n^{3(k+2)})$ in the number of vertices $n$. Our algorithm applies, for example, to Vietoris--Rips complexes of points sampled from a curve in $\mathbb{R}^d$ when the scale is bounded depending on…

使用持久同调检测动力系统的不变流形

Detecting invariant manifolds of dynamical systems using persistent homology

该研究运用持久同调理论方法，先建立接近集合持久图的联系，再通过数值样本集的持久图研究特定动力系统极限集性质，在温和假设下借助数值数据得出极限集几何的分析结果。