Publications [#226429] of Paul L Bendich

Papers Accepted

1. Liz Munch, Paul Bendich, Kate Turner, Sayan Mukherjee, Jonathan Mattingly, and John Harer, Probabalistic Frechet Means and Statistics on Vineyards, Electronic Journal of Statistics (2015) [6530]
   (last updated on 2015/04/28)

   Author's Comments:
   to appear

   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. In [21], Mileyko and his collaborators made the rst study of the properties of the Frechet mean in (Dp;Wp), the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Frechet mean of a nite 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 Frechet mean denition to the realm of vineyards. We x this problem by altering the original denition 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 the (Frechet mean) persistence diagram of a perturbation of the input diagrams. We show that this new denition denes a (Holder) 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.