Probabilistic Fréchet means for time varying persistence diagrams
Abstract
© 2015, Institute of Mathematical Statistics. All rights reserved.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 [23], Mileyko and his collaborators
made the first study of the properties of the Fréchet mean in (D<inf>p</inf>, W<inf>p</inf>),
the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular,
they showed that the Fréchet mean of a finite set of diagrams always exists, but is
not necessarily unique. The means of a continuously-varying set of diagrams do not
themselves (necessarily) vary continuously, which presents obvious problems when trying
to extend the Fréchet mean definition to the realm of time-varying persistence diagrams,
better known as vineyards. We fix this problem by altering the original definition
of Fréchet 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 (D<inf>p</inf>)N→ℙ(D<inf>p</inf>). We show that this map is Hölder continuous on finite diagrams and
thus can be used to build a useful statistic on vineyards.
Type
Journal articlePermalink
https://hdl.handle.net/10161/10051Published Version (Please cite this version)
10.1214/15-EJS1030Publication Info
Munch, Elizabeth; Bendich, Paul; Turner, Katharine; Mukherjee, Sayan; Mattingly, Jonathan;
& Harer, John (2015). Probabilistic Fréchet means for time varying persistence diagrams. Electronic Journal of Statistics, 9. pp. 1173-1204. 10.1214/15-EJS1030. Retrieved from https://hdl.handle.net/10161/10051.This is constructed from limited available data and may be imprecise. To cite this
article, please review & use the official citation provided by the journal.
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Show full item recordScholars@Duke
Paul L Bendich
Associate Research Professor of Mathematics
I am a mathematician whose main research focus lies in adapting theory from ostensibly
pure areas of mathematics, such as topology, geometry, and abstract algebra, into
tools that can be broadly used in many data-centeredapplications.My initial training
was in a recently-emerging field called topological data analysis (TDA). I have beenresponsible
for several essential and widely-used elements of its theoretical toolkit, with a
particularfocus on building TDA methodology
John Harer
Professor Emeritus of Mathematics
Professor Harer's primary research is in the use of geometric, combinatorial and computational
techniques to study a variety of problems in data analysis, shape recognition, image
segmentation, tracking, cyber security, ioT, biological networks and gene expression.
Jonathan Christopher Mattingly
James B. Duke Distinguished Professor
Jonathan Christopher Mattingly grew up in Charlotte, NC where he attended Irwin Ave
elementary and Charlotte Country Day. He graduated from the NC School of Science
and Mathematics and received a BS is Applied Mathematics with a concentration in physics
from Yale University. After two years abroad with a year spent at ENS Lyon studying
nonlinear and statistical physics on a Rotary Fellowship, he returned to the US to
attend Princeton University where he obtained a PhD in Applied and
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