Math @ Duke

Publications [#226628] of Paul L Bendich
Papers Published
 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. 11731204 [6530]
(last updated on 2015/06/27)
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 pth 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 continuouslyvarying set of diagrams do not themselves vary continuously, which
presents obvious problems when trying to extend the Frechet mean denition to the realm of vineyards.
We x this problem by altering the original denition 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 denition denes 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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