Probabilistic Fréchet means for time varying persistence diagrams
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2015-01-01
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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 [23], Mileyko and his collaborators made the first study of the properties of the Fréchet mean in (Dp, Wp), 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 (Dp)N→ℙ(Dp). We show that this map is Hölder continuous on finite diagrams and thus can be used to build a useful statistic on vineyards.
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Munch, Elizabeth, Paul Bendich, Katharine Turner, Sayan Mukherjee, Jonathan Mattingly and John Harer (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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Paul L Bendich
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-centered applications.
My initial training was in a recently-emerging field called topological data analysis (TDA). I have been responsible for several essential and widely-used elements of its theoretical toolkit, with a particular focus on building TDA methodology for use on stratified spaces. Some of this work involves the creation of efficient algorithms, but much of it centers around theorem-proof mathematics, using proof techniques not only from algebraic topology, but also from computational geometry, from probability, and from abstract algebra.
Recently, I have done foundational work on TDA applications in several areas, including to neuroscience, to multi-target tracking, to multi-modal data fusion, and to a probabilistic theory of database merging. I am also becoming involved in efforts to integrate TDA within deep learning theory and practice.
I typically teach courses that connect mathematical principles to machine learning, including upper-level undergraduate courses in topological data analysis and more general high-dimensional data analysis, as well as a sophomore level course (joint between pratt and math) that serves as a broad introduction to machine learning and data analysis concepts.
Jonathan Christopher Mattingly
Jonathan Christopher Mattingly grew up in Charlotte, NC, where he attended Irwin Avenue 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 Computational Mathematics in 1998. After 4 years as a Szego assistant professor at Stanford University and a year as a member of the IAS in Princeton, he moved to Duke in 2003. He is currently a professor of mathematics and statistical science.
His expertise is in the longtime behavior of stochastic system including randomly forced fluid dynamics, turbulence, stochastic algorithms used in molecular dynamics and Bayesian sampling, and stochasticity in biochemical networks.
Since 2013 he has also been working to understand and quantify gerrymandering and its interaction of a region's geopolitical landscape. This has lead him to testify in a number of court cases including in North Carolina, which led to the NC congressional and both NC legislative maps being deemed unconstitutional and replaced for the 2020 elections.
He is the recipient of a Sloan Fellowship and a PECASE CAREER award. He is also a fellow of the IMS, the AMS, SIAM and AAAS. He was awarded the Defender of Freedom award by Common Cause for his work on Quantifying Gerrymandering.
John Harer
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.
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