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.
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 .
@cite_27 @cite_42 @cite_39
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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 [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. A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first homology. We call the resulting object persistence. Recent results demonstrate that persistence of a compact geodesic locally contractible space X carries a lot of geometric information. However, by definition the corresponding Rips complexes have uncountably many vertices. In this paper we show that nonetheless, the whole persistence of X may be obtained by an appropriate finite sample (subset of X), and that persistence of any subset of X is well interleaved with the persistence of X. It follows that the persistence of X is the minimum of persistences obtained by all finite samples. Furthermore, we prove a much improved Stability theorem for such approximations. As a special case we provide for each (r>0 ) a density (s>0 ), so that for each s-dense sample (S X ) the corresponding fundamental group (and the first homology) of the Rips complex of S is isomorphic to the one of X, leading to an improved reconstruction result. Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane @math that can equivalently be seen as discrete measures in @math . When the data come as a random point cloud, these discrete measures become random measures whose expectation is studied in this paper. First, we show that for a wide class of filtrations, including the C ech and Rips-Vietoris filtrations, the expected persistence diagram, that is a deterministic measure on @math , has a density with respect to the Lebesgue measure. Second, building on the previous result we show that the persistence surface recently introduced in [Adams & al., Persistence images: a stable vector representation of persistent homology] can be seen as a kernel estimator of this density. We propose a cross-validation scheme for selecting an optimal bandwidth, which is proven to be a consistent procedure to estimate the density. We study barycenters in the space of probability measures on a Riemannian manifold, equipped with the Wasserstein metric. Under reasonable assumptions, we establish absolute continuity of the barycenter of general measures Ω∈P(P(M))Ω∈P(P(M)) on Wasserstein space, extending on one hand, results in the Euclidean case (for barycenters between finitely many measures) of Agueh and Carlier [1] to the Riemannian setting, and on the other hand, results in the Riemannian case of Cordero-Erausquin, McCann, Schmuckenschlager [12] for barycenters between two measures to the multi-marginal setting. Our work also extends these results to the case where Ω is not finitely supported. As applications, we prove versions of Jensen's inequality on Wasserstein space and a generalized Brunn–Minkowski inequality for a random measurable set on a Riemannian manifold.