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diff --git a/src/python/doc/barycenter_sum.inc b/src/python/doc/barycenter_sum.inc new file mode 100644 index 00000000..da2bdd84 --- /dev/null +++ b/src/python/doc/barycenter_sum.inc @@ -0,0 +1,24 @@ +.. table:: + :widths: 30 50 20 + + +-----------------------------------------------------------------+----------------------------------------------------------------------+------------------------------------------------------------------+ + | .. 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). | | + +-----------------------------------------------------------------+----------------------------------------------------------------------+------------------------------------------------------------------+ + | * :doc:`barycenter_user` | | + +-----------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------+ |