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Wasserstein distance user manual
================================
Definition
----------

.. include:: wasserstein_distance_sum.inc

This implementation is based on ideas from "Frechet means for distribution of persistence diagrams", Turner et al. 2014.

Function
--------
.. autofunction:: gudhi.barycenter.lagrangian_barycenter


Basic example
-------------

This example computes the Frechet mean (aka Wasserstein barycenter) between four persistence diagrams.
It is initialized on the 4th diagram, which is the empty diagram. It is encoded by np.array([]).
Note that persistence diagrams must be submitted as (n x 2) numpy arrays and must not contain inf values.

.. testcode::

    import gudhi.barycenter
    import numpy as np

    dg1 = np.array([[0.2, 0.5]])
    dg2 = np.array([[0.2, 0.7]])
    dg3 = np.array([[0.3, 0.6], [0.7, 0.8], [0.2, 0.3]])
    dg4 = np.array([])
    
    bary = gudhi.barycenter.lagrangian_barycenter(pdiagset=[dg1, dg2, dg3, dg4],init=3))

    message = "Wasserstein barycenter estimated:"    
    print(message)
    print(bary)

The output is:

.. testoutput::

    Wasserstein barycenter estimated:
    [[0.27916667 0.55416667]
     [0.7375     0.7625    ]
     [0.2375     0.2625    ]]