Geometric embeddings of spaces of persistence diagrams with explicit distortions

Abstract

Let $n$ be a positive integer. We provide an explicit geometrically motivated $1$-Lipschitz map from the space of persistence diagrams on $n$ points (equipped with the Bottleneck distance) into the Hilbert space $\ell^2$. Such maps are a crucial step in topological data analysis, allowing the use of statistical methods (and thus data analysis) on collections of persistence diagrams. The main advantage of our maps as compared to most of the other such vectorizations is that they are coarse and uniform embeddings with explicit distortion functions. This allows us to control the amount of geometric information lost through their application. Furthermore, we also provide an explicit $1$-Lipschitz map from the space of persistence diagrams on $n$ points on a bounded domain into a Euclidean space with an explicit distortion function. We conclude with a differently flavored embedding of the space of persistence diagrams on $n$ points on a bounded domain into $\mathbb{R}^{n(n+1)}$. The maps we construct are fairly simple, with each component depending only on the bottleneck distance to the corresponding ``landmark" persistence diagram. Due to geometric motivation from classical dimension theory, our methods are best described as quantitative dimension theory.