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Diffstat (limited to 'src/python/doc')
| -rw-r--r-- | src/python/doc/barycenter_sum.inc | 24 | ||||
| -rw-r--r-- | src/python/doc/barycenter_user.rst | 60 | ||||
| -rw-r--r-- | src/python/doc/img/barycenter.png | bin | 0 -> 12433 bytes | |||
| -rw-r--r-- | src/python/doc/index.rst | 5 |
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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` | | + +-----------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------+ diff --git a/src/python/doc/barycenter_user.rst b/src/python/doc/barycenter_user.rst new file mode 100644 index 00000000..83e9bebb --- /dev/null +++ b/src/python/doc/barycenter_user.rst @@ -0,0 +1,60 @@ +:orphan: + +.. To get rid of WARNING: document isn't included in any toctree + +Barycenter user manual +================================ +Definition +---------- + +.. include:: barycenter_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. +As the algorithm is not convex, its output depends on the initialization and +is only a local minimum of the objective function. +Initialization can be either given as an integer (in which case the i-th +diagram of the list is used as initial estimate) or as a diagram. +If None, it will randomly select one of the diagram of the list +as initial estimate. +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([]) + pdiagset = [dg1, dg2, dg3, dg4] + bary = gudhi.barycenter.lagrangian_barycenter(pdiagset=pdiagset,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 ]] + + diff --git a/src/python/doc/img/barycenter.png b/src/python/doc/img/barycenter.png Binary files differnew file mode 100644 index 00000000..cad6af70 --- /dev/null +++ b/src/python/doc/img/barycenter.png diff --git a/src/python/doc/index.rst b/src/python/doc/index.rst index 3387a64f..96cd3513 100644 --- a/src/python/doc/index.rst +++ b/src/python/doc/index.rst @@ -71,6 +71,11 @@ Wasserstein distance .. include:: wasserstein_distance_sum.inc +Barycenter +============ + +.. include:: barycenter_sum.inc + Persistence representations =========================== |
