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Publications [#258517] of Sayan Mukherjee

Papers Published

  1. Munch, E; Turner, K; Bendich, P; Mukherjee, S; Mattingly, J; Harer, J, Probabilistic Fréchet means for time varying persistence diagrams, Electronic Journal of Statistics, vol. 9 no. 1 (January, 2015), pp. 1173-1204 [repository], [doi]
    (last updated on 2017/03/24)

    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 (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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