Estimation and Quantization of Expected Persistence Diagrams

Persistence diagrams (PDs) are the most common descriptors used to encode the topology of structured data appearing in challenging learning tasks;~think e.g.~of graphs, time series or point clouds sampled close to a manifold. Given random objects and the corresponding distribution of PDs, one may want to build a statistical summary---such as a mean---of these random PDs, which is however not a trivial task as the natural geometry of the space of PDs is not linear. In this article, we study two such summaries, the Expected Persistence Diagram (EPD), and its quantization. The EPD is a measure supported on $\mathbb{R}^2$, which may be approximated by its empirical counterpart. We prove that this estimator is optimal from a minimax standpoint on a large class of models with a parametric rate of convergence. The empirical EPD is simple and efficient to compute, but possibly has a very large support, hindering its use in practice. To overcome this issue, we propose an algorithm to compute a quantization of the empirical EPD, a measure with small support which is shown to approximate with near-optimal rates a quantization of the theoretical EPD.