Math @ Duke

Publications [#221260] of Jonathan C. Mattingly
Papers Submitted
 Elizabeth Munch, Paul Bendich, Katharine Turner, Sayan Mukherjee, Jonathan Mattingly, John Harer, Probabilistic Fréchet Means and Statistics on Vineyards
(2013) [arXiv:1307.6530]
(last updated on 2013/12/22)
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 pth 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 continuouslyvarying 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.


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