## Fréchet Means for Distributions of Persistence Diagrams

Publication
, Journal Article

Turner, K; Mileyko, Y; Mukherjee, S; Harer, J

Published in: Discrete and Computational Geometry

January 1, 2014

Given a distribution ρ on persistence diagrams and observations (Formula presented.) 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 (Formula presented.) then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fréchet mean computed by the algorithm given observations drawn iid from ρ. We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields. © 2014 Springer Science+Business Media New York.

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## Published In

Discrete and Computational Geometry

## DOI

## EISSN

1432-0444

## ISSN

0179-5376

## Publication Date

January 1, 2014

## Volume

52

## Issue

1

## Start / End Page

44 / 70

## Related Subject Headings

- Computation Theory & Mathematics
- 49 Mathematical sciences
- 46 Information and computing sciences
- 0802 Computation Theory and Mathematics
- 0103 Numerical and Computational Mathematics
- 0101 Pure Mathematics

### Citation

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Turner, K., Mileyko, Y., Mukherjee, S., & Harer, J. (2014). Fréchet Means for Distributions of Persistence Diagrams.

*Discrete and Computational Geometry*,*52*(1), 44–70. https://doi.org/10.1007/s00454-014-9604-7Turner, K., Y. Mileyko, S. Mukherjee, and J. Harer. “Fréchet Means for Distributions of Persistence Diagrams.”

*Discrete and Computational Geometry*52, no. 1 (January 1, 2014): 44–70. https://doi.org/10.1007/s00454-014-9604-7.Turner K, Mileyko Y, Mukherjee S, Harer J. Fréchet Means for Distributions of Persistence Diagrams. Discrete and Computational Geometry. 2014 Jan 1;52(1):44–70.

Turner, K., et al. “Fréchet Means for Distributions of Persistence Diagrams.”

*Discrete and Computational Geometry*, vol. 52, no. 1, Jan. 2014, pp. 44–70.*Scopus*, doi:10.1007/s00454-014-9604-7.Turner K, Mileyko Y, Mukherjee S, Harer J. Fréchet Means for Distributions of Persistence Diagrams. Discrete and Computational Geometry. 2014 Jan 1;52(1):44–70.

## Published In

Discrete and Computational Geometry

## DOI

## EISSN

1432-0444

## ISSN

0179-5376

## Publication Date

January 1, 2014

## Volume

52

## Issue

1

## Start / End Page

44 / 70

## Related Subject Headings

- Computation Theory & Mathematics
- 49 Mathematical sciences
- 46 Information and computing sciences
- 0802 Computation Theory and Mathematics
- 0103 Numerical and Computational Mathematics
- 0101 Pure Mathematics