**Papers Published**

- Elizabeth Munch, Paul Bendich, Katharine Turner, Sayan Mukherjee, Jonathan Mattingly, John Harer,
*Probabilistic Fréchet Means and Statistics on Vineyards*(2014) (http://arxiv.org/abs/1307.6530.)

(last updated on 2014/12/15)**Abstract:**

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. Mileyko and his collaborators made the first study of the properties of the Fr\'{e}chet mean in (Dp,Wp), the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Fr\'{e}chet mean of a finite set of diagrams always exists, but is not necessarily unique. As an unfortunate consequence, one sees that the means of a continuously-varying set of diagrams do not themselves vary continuously, which presents obvious problems when trying to extend the Fr\'{e}chet mean definition to the realm of vineyards. We fix this problem by altering the original definition of Fr\'{e}chet 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 the (Fr\'{e}chet mean) persistence diagram of a perturbation of the input diagrams. We show that this new definition defines a (H\"older) continuous map, for each k, from (Dp)k→P(Dp), and we present several examples to show how it may become a useful statistic on vineyards.