Math @ Duke

Publications [#225825] of John Harer
Papers Accepted
 E. Munch, P. Bendich, K. Turner, S. Mukherjee, J. Mattingly and J. Harer, Probabilistic Frechet Means and Statistics on Vineyards,
Foundations of Computational Math
(2014)
(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. In [21], Mileyko and his collaborators made the first study of the properties of the Fr ́echet mean in (Dp,Wp), the space of persistence diagrams equipped with the pth Wasserstein metric. In particular, they showed that the Fr ́echet 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 ́echet mean definition to the realm of vineyards.
We fix this problem by altering the original definition of Fr ́echet 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 ́echet 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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