Topological data analysis provides a multiscale description of the geometry and topology of quantitative data. The persistence landscape is a topological summary that can be easily combined with tools from statistics and machine learning. We give efficient algorithms for calculating persistence landscapes, their averages, and distances between such averages. We discuss an implementation of these algorithms and some related procedures. These are intended to facilitate the combination of statistics and machine learning with topological data analysis. We present an experiment showing that the low-dimensional persistence landscapes of points sampled from spheres (and boxes) of varying dimensions differ.
@cite_27: 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 [20], Mileyko and his collaborators made the first study of the properties of the Frechet mean in (Dp,Wp), 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. As an unfortunate consequence, one sees that the means of a continuously-varying set of diagrams do not themselves 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 the (Frechet mean) persistence diagram of a perturbation of the input diagrams. We show that this new definition defines a (Holder) continuous map, for each k, from (Dp) k → P (Dp), and we present several examples to show how it may become a useful statistic on vineyards.
@cite_42: Given a distribution @math ? on persistence diagrams and observations @math X 1 , ? , X n i i d ? we introduce an algorithm in this paper that estimates a Frechet mean from the set of diagrams @math X 1 , ? , X n . If the underlying measure @math ? is a combination of Dirac masses @math ? = 1 m ? i = 1 m ? Z i then we prove the algorithm converges to a local minimum and a law of large numbers result for a Frechet mean computed by the algorithm given observations drawn iid from @math ? . We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.
@cite_39: We prove two stability results for Lipschitz functions on triangulable, compact metric spaces and consider applications of both to problems in systems biology. Given two functions, the first result is formulated in terms of the Wasserstein distance between their persistence diagrams and the second in terms of their total persistence.
It has been shown @cite_39 that the set of persistence diagrams with the Wasserstein distance is a complete and separable metric space, and thus provides a suitable setting for probability and statistics. Unfortunately the Fr 'echet mean is not necessarily unique. For a slightly adjusted metric, there is an algorithm @cite_42 that converges to an element of the Fr 'echet mean set, though it does not have good computational properties. The discontinuity of this procedure can be remedied by using a probabilistic approach @cite_27 .