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Register a new userProbabilistic Frechet Means for Time Varying Persistence Diagrams
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Added by
Elizabeth Munch
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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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 Frechet mean in $(mathcal{D}_p,W_p)$, the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Frechet 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 Frechet mean definition to the realm of vineyards. We fix this problem by altering the original definition of Frechet 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 Holder continuous on finite diagrams and thus can be used to build a useful statistic on time-varying persistence diagrams, better known as vineyards.
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For $1 le p < infty$, the Frechet $p$-mean of a probability distribution $mu$ on a metric space $(X,d)$ is the set $F_p(mu) := {arg,min}_{xin X}int_{X}d^p(x,y), dmu(y)$, which is taken to be empty if no minimizer exists. Given a sequence $(Y_i)_{i in mathbb{N}}$ of independent, identically distributed random samples from some probability measure $mu$ on $X$, the Frechet $p$-means of the empirical measures, $F_p(frac{1}{n}sum_{i=1}^{n}delta_{Y_i})$ form a sequence of random closed subsets of $X$. We investigate the senses in which this sequence of random closed sets and related objects converge almost surely as $n to infty$.
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