Geometric embeddings of spaces of persistence diagrams with explicit distortions

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 Hilbert space. Such maps are acrucial step in topological data analysis, allowing the use of statistic (andthus data analysis) on collections of persistence diagrams. The main advantageof our maps as compared to most of the other such transformations is that theyare coarse and uniform embeddings with explicit distortion functions. Thisallows us to control the amount of geometric information lost through theirapplication. Furthermore, we provide an explicit $1$-Lipschitz map from thespace of persistence diagrams on $n$ points on a bounded domain into aEuclidean space with an explicit distortion function. The mentioned maps arefairly simple, with each component depending depending only on the bottleneckdistance to the corresponding landmark persistence diagram. Due to geometricmotivation from classical dimension theory, our methods are best described asquantitative dimension theory. We discuss the advantages and disadvantages ofour approach. We conclude with differently flavoured embedding of the space ofpersistence diagrams on $n$ points on a bounded domain into$\mathbb{R}^{n(n+1)}$.