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Publications [#226628] of Paul L Bendich

Papers Published

  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, vol. 9 (2015), pp. 1173-1204 [6530]
    (last updated on 2015/06/27)

    Author's Comments:
    to appear

    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.
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