Probabilistic Fréchet means for time varying persistence diagrams
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© 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 , 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.
Published Version (Please cite this version)10.1214/15-EJS1030
Publication InfoMunch, 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.
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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
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.
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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