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author | tlacombe <lacombe1993@gmail.com> | 2019-12-05 18:42:48 +0100 |
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committer | tlacombe <lacombe1993@gmail.com> | 2019-12-05 18:42:48 +0100 |
commit | 80aa14d1b92d1a61366d798b07073289d4db4fda (patch) | |
tree | d6fbe5c7bc404e91517154b3067f414f39bcb820 /src/python/doc | |
parent | 48f7e17c5e9d4f6936bfdf6384015fe833e30c74 (diff) |
first version of barycenter for persistence diagrams
Diffstat (limited to 'src/python/doc')
-rw-r--r-- | src/python/doc/barycenter_sum.inc | 22 | ||||
-rw-r--r-- | src/python/doc/barycenter_user.rst | 51 |
2 files changed, 73 insertions, 0 deletions
diff --git a/src/python/doc/barycenter_sum.inc b/src/python/doc/barycenter_sum.inc new file mode 100644 index 00000000..7801a845 --- /dev/null +++ b/src/python/doc/barycenter_sum.inc @@ -0,0 +1,22 @@ +.. table:: + :widths: 30 50 20 + + +-----------------------------------------------------------------+----------------------------------------------------------------------+------------------------------------------------------------------+ + | .. figure:: | A Frechet mean (or barycenter) is a generalization of the arithmetic | :Author: Theo Lacombe | + | ../../doc/Barycenter/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 persis- | | + | | tence diagrams. | | + | | It is known to exist and is generically unique. However, an exact | | + | | computation is in general untractable. Current implementation avai- | :Requires: Python Optimal Transport (POT) :math:`\geq` 0.5.1 | + | | -lable 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 combinatorial structure of the algorithm limits its | | + | | scaling on large scale problems (thousands of diagrams and of points | | + | | per diagram). | | + +-----------------------------------------------------------------+----------------------------------------------------------------------+------------------------------------------------------------------+ + | * :doc:`barycenter_user` | | + +-----------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------+ diff --git a/src/python/doc/barycenter_user.rst b/src/python/doc/barycenter_user.rst new file mode 100644 index 00000000..fae2854a --- /dev/null +++ b/src/python/doc/barycenter_user.rst @@ -0,0 +1,51 @@ +:orphan: + +.. To get rid of WARNING: document isn't included in any toctree + +Wasserstein distance user manual +================================ +Definition +---------- + +.. include:: wasserstein_distance_sum.inc + +This implementation is based on ideas from "Large Scale Computation of Means and Cluster for Persistence Diagrams via Optimal Transport". + +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 ]] + + |