# Fréchet Means for Distributions of Persistence Diagrams

@article{Turner2014FrchetMF, title={Fr{\'e}chet Means for Distributions of Persistence Diagrams}, author={Katharine Turner and Yuriy Mileyko and Sayan Mukherjee and John Harer}, journal={Discrete \& Computational Geometry}, year={2014}, volume={52}, pages={44-70} }

Given a distribution $$\rho $$ρ on persistence diagrams and observations $$X_{1},\ldots ,X_{n} \mathop {\sim }\limits ^{iid} \rho $$X1,…,Xn∼iidρ we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams $$X_{1},\ldots ,X_{n}$$X1,…,Xn. If the underlying measure $$\rho $$ρ is a combination of Dirac masses $$\rho = \frac{1}{m} \sum _{i=1}^{m} \delta _{Z_{i}}$$ρ=1m∑i=1mδZi then we prove the algorithm converges to a local minimum and a law of large numbers result…

## 192 Citations

The density of expected persistence diagrams and its kernel based estimation

- Mathematics, Computer ScienceSoCG
- 2018

A cross-validation scheme for selecting an optimal bandwidth is proposed, which is proven to be a consistent procedure to estimate the density.

Probabilistic Fréchet means for time varying persistence diagrams

- Mathematics
- 2015

This work alters the original definition of Fr\'echet mean so that it now becomes a probability measure on the set of persistence diagrams and shows 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.

Persistent Homology Transform for Modeling Shapes and Surfaces

- Mathematics
- 2013

In this paper we introduce a statistic, the persistent homology transform (PHT), to model surfaces in $\mathbb{R}^3$ and shapes in $\mathbb{R}^2$. This statistic is a collection of persistence…

A smeary central limit theorem for manifolds with application to high-dimensional spheres

- Mathematics
- 2018

The (CLT) central limit theorems for generalized Frechet means (data descriptors assuming values in stratified spaces, such as intrinsic means, geodesics, etc.) on manifolds from the literature are…

$\zeta$-functions and the topology of superlevel sets of stochastic processes

- Mathematics
- 2021

We describe the topology of superlevel sets of (α-stable) Lévy processes X by introducing so-called stochastic ζ-functions, which are defined in terms of the widely used Perspfunctional in the theory…

Probabilistic Fréchet Means and Statistics on Vineyards

- MathematicsArXiv
- 2013

This work alters the original definition of Frechet mean so that it now becomes a probability measure on the set of persistence diagrams, where each atom is itself the (Frechet mean) persistence diagram of a perturbation of the input diagrams.

Convergence of persistence diagram in the sparse regime

- Mathematics
- 2021

The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Čech filtration. A persistence diagram is a graphical descriptor of a topological and…

CONVERGENCE OF PERSISTENCE DIAGRAM IN THE SUBCRITICAL REGIME

- Mathematics
- 2021

The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Čech filtration. A persistence diagram is a graphical descriptor of a topological and…

Strong laws of large numbers for Fr\'echet means

- Mathematics
- 2020

For 1 ≤ p < ∞, the Fréchet p-mean of a probability distribution μ on a metric space (X, d) is the set Fp(μ) := arg minx∈X ∫ X d(x, y) dμ(y), which is taken to be empty if no minimizer exists. Given a…

Means and medians of sets of persistence diagrams

- MathematicsArXiv
- 2013

The space of persistence diagrams is looked at under a variety of different metrics which are analogous to L p metrics on the space of functions, which gives the natural definitions of both the mean and median of a finite number of persistence diagram.

## References

SHOWING 1-10 OF 29 REFERENCES

Statistical topology via Morse theory, persistence and nonparametric estimation

- Mathematics
- 2009

In this paper we examine the use of topological methods for multivariate statistics. Using persistent homology from computational algebraic topology, a random sample is used to construct estimators…

Probability measures on the space of persistence diagrams

- Mathematics
- 2011

This paper shows that the space of persistence diagrams has properties that allow for the definition of probability measures which support expectations, variances, percentiles and conditional…

A Topological View of Unsupervised Learning from Noisy Data

- Computer Science, MathematicsSIAM J. Comput.
- 2011

It is shown that if the variance of the Gaussian noise is small in a certain sense, then the homology can be learned with high confidence by an algorithm that has a weak (linear) dependence on the ambient dimension.

Ricci curvature for metric-measure spaces via optimal transport

- Mathematics
- 2004

We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the…

Central limit theorems for some graphs in computational geometry

- Mathematics
- 2001

Let Bn be an increasing sequence of regions in d-dimensional space with volume n and with union d. We prove a general central limit theorem for functionals of point sets, obtained either by…

Probability Measures on Metric Spaces of Nonpositive Curvature

- Mathematics
- 2003

We present an introduction to metric spaces of nonpositive curvature (”NPC spaces”) and a discussion of barycenters of probability measures on such spaces. In our introduction to NPC spaces, we will…

Geometric Representations of Hypergraphs for Prior Specification and Posterior Sampling

- Mathematics, Computer Science
- 2009

This parametrization of hypergraphs based on the geometry of points in R can recover both the junction tree factorization as well as the hyper Markov law and is used to infer conditional independence models or Markov structure of multivariate distributions.

Random Geometric Complexes

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 2011

The expected topological properties of Čech and Vietoris–Rips complexes built on random points in ℝd are studied and asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes are given.

Local homology transfer and stratification learning

- Mathematics, Computer ScienceSODA
- 2012

This paper uses methods derived from kernel and cokernel persistent homology to cluster the data points into different strata, and provides a probabilistic guarantee for the clustering for the point sample setting.