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.. table::
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| .. figure:: | A Frechet mean (or barycenter) is a generalization of the arithmetic | :Author: Theo Lacombe |
| ./img/barycenter.png | mean in a non linear space such as the one of persistence diagrams. | |
| :figclass: align-center | Given a set of persistence diagrams :math:`\mu_1 \dots \mu_n`, it is | :Introduced in: GUDHI 3.1.0 |
| | defined as a minimizer of the variance functional, that is of | |
| Illustration of Frechet mean between persistence | :math:`\mu \mapsto \sum_{i=1}^n d_2(\mu,\mu_i)^2`. | :Copyright: MIT |
| diagrams. | where :math:`d_2` denotes the Wasserstein-2 distance between | |
| | persistence diagrams. | |
| | It is known to exist and is generically unique. However, an exact | |
| | computation is in general untractable. Current implementation | :Requires: Python Optimal Transport (POT) :math:`\geq` 0.5.1 |
| | available is based on [Turner et al, 2014], and uses an EM-scheme to | |
| | provide a local minimum of the variance functional (somewhat similar | |
| | to the Lloyd algorithm to estimate a solution to the k-means | |
| | problem). The local minimum returned depends on the initialization of| |
| | the barycenter. | |
| | The combinatorial structure of the algorithm limits its | |
| | scaling on large scale problems (thousands of diagrams and of points | |
| | per diagram). | |
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| * :doc:`barycenter_user` | |
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