Math @ Duke

Publications [#303522] of Paul L Bendich
Papers Published
 Munch, E; Turner, K; Bendich, P; Mukherjee, S; Mattingly, J; Harer, J, Probabilistic Fréchet Means for Time Varying Persistence Diagrams, vol. 9 no. 1
(January, 2015),
pp. 11731204 [1307.6530v3], [doi]
(last updated on 2018/10/18)
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 2011, Mileyko and his collaborators made the first study of the properties
of the Fr\'echet mean in $(\mathcal{D}_p,W_p)$, 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. The means of a continuouslyvarying set of diagrams do not
themselves (necessarily) 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 a persistence diagram determined using a
perturbation of the input diagrams. This definition gives for each $N$ a map
$(\mathcal{D}_p)^N \to \mathbb{P}(\mathcal{D}_p)$. We show that this map is
H\"older continuous on finite diagrams and thus can be used to build a useful
statistic on timevarying persistence diagrams, better known as vineyards.


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