Probabilistic Fréchet means for time varying persistence diagrams

Probabilistic Fr\'echet Means for Time Varying Persistence Diagrams

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 [21], 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 f...

Fréchet Means for Distributions of Persistence Diagrams

Given a distribution ρ on persistence diagrams and observations X1, ...Xn iid ∼ ρ we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams X1, ...Xn. If the underlying measure ρ is a combination of Dirac masses ρ = 1 m ∑m i=1 δZi then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fréchet mean computed by the alg...

Probabilistic Fréchet Means and Statistics on Vineyards

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 [21], 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 f...

Fréchet Means of Curves for Signal

Means and medians of sets of persistence diagrams

The persistence diagram is the fundamental object in topological data analysis. It inherits the stochastic variability of the data we use as input. As such we need to understand how to perform statistics on the space of persistence diagrams. This paper looks at the space of persistence diagrams under a variety of different metrics which are analogous to L metrics on the space of functions. Usin...