Probabilistic Fr\'echet Means for Time Varying Persistence Diagrams

In order to use persistence diagrams as a true statistical tool, it would be very useful to have a good notion of mean and variance for a set of diagrams. In 2011, Mileyko and his collaborators made the first study of the properties of the Fr\'echet mean in $(\mathcal{D}_p,W_p)$, the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Fr\'echet mean of a finite set of diagrams always exists, but is not necessarily unique. The means of a continuously-varying set of diagrams do not themselves (necessarily) vary continuously, which presents obvious problems when trying to extend the Fr\'echet mean definition to the realm of vineyards. We fix this problem by altering the original definition of Fr\'echet mean so that it now becomes a probability measure on the set of persistence diagrams; in a nutshell, the mean of a set of diagrams will be a weighted sum of atomic measures, where each atom is itself a persistence diagram determined using a perturbation of the input diagrams. This definition gives for each $N$ a map $(\mathcal{D}_p)^N \to \mathbb{P}(\mathcal{D}_p)$. We show that this map is H\"older continuous on finite diagrams and thus can be used to build a useful statistic on time-varying persistence diagrams, better known as vineyards.