Probabilistic Frechet 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 [23], Mileyko and his collaborators made the first study of the properties of the Frechet mean in (D-p, W-p), the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Frechet 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 Frechet mean definition to the realm of time-varying persistence diagrams, better known as vineyards. We fix this problem by altering the original definition of Frechet 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 (D-p)(N) -> P(D-p). We show that this map is Holder continuous on finite diagrams and thus can be used to build a useful statistic on vineyards.