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Diffstat (limited to 'src/python/doc/wasserstein_distance_user.rst')
| -rw-r--r-- | src/python/doc/wasserstein_distance_user.rst | 96 |
1 files changed, 88 insertions, 8 deletions
diff --git a/src/python/doc/wasserstein_distance_user.rst b/src/python/doc/wasserstein_distance_user.rst index a9b21fa5..c443bab5 100644 --- a/src/python/doc/wasserstein_distance_user.rst +++ b/src/python/doc/wasserstein_distance_user.rst @@ -9,10 +9,16 @@ Definition .. include:: wasserstein_distance_sum.inc -Functions ---------- -This implementation uses the Python Optimal Transport library and is based on -ideas from "Large Scale Computation of Means and Cluster for Persistence +The q-Wasserstein distance is defined as the minimal value achieved +by a perfect matching between the points of the two diagrams (+ all +diagonal points), where the value of a matching is defined as the +q-th root of the sum of all edge lengths to the power q. Edge lengths +are measured in norm p, for :math:`1 \leq p \leq \infty`. + +Distance Functions +------------------ +This first implementation uses the Python Optimal Transport library and is based +on ideas from "Large Scale Computation of Means and Cluster for Persistence Diagrams via Optimal Transport" :cite:`10.5555/3327546.3327645`. .. autofunction:: gudhi.wasserstein.wasserstein_distance @@ -26,7 +32,7 @@ Morozov, and Arnur Nigmetov. .. autofunction:: gudhi.hera.wasserstein_distance Basic example -------------- +************* This example computes the 1-Wasserstein distance from 2 persistence diagrams with Euclidean ground metric. Note that persistence diagrams must be submitted as (n x 2) numpy arrays and must not contain inf values. @@ -48,9 +54,9 @@ The output is: Wasserstein distance value = 1.45 -We can also have access to the optimal matching by letting `matching=True`. +We can also have access to the optimal matching by letting `matching=True`. It is encoded as a list of indices (i,j), meaning that the i-th point in X -is mapped to the j-th point in Y. +is mapped to the j-th point in Y. An index of -1 represents the diagonal. .. testcode:: @@ -78,9 +84,83 @@ An index of -1 represents the diagonal. The output is: .. testoutput:: - + Wasserstein distance value = 2.15 point 0 in dgm1 is matched to point 0 in dgm2 point 1 in dgm1 is matched to point 2 in dgm2 point 2 in dgm1 is matched to the diagonal point 1 in dgm2 is matched to the diagonal + +Barycenters +----------- + +A Frechet mean (or barycenter) is a generalization of the arithmetic +mean in a non linear space such as the one of persistence diagrams. +Given a set of persistence diagrams :math:`\mu_1 \dots \mu_n`, it is +defined as a minimizer of the variance functional, that is of +:math:`\mu \mapsto \sum_{i=1}^n d_2(\mu,\mu_i)^2`. +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 +available is based on (Turner et al., 2014), +:cite:`turner2014frechet` +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 +performances on large scale problems (thousands of diagrams and of points +per diagram). + +.. figure:: + ./img/barycenter.png + :figclass: align-center + + Illustration of Frechet mean between persistence + diagrams. + + +.. autofunction:: gudhi.wasserstein.barycenter.lagrangian_barycenter + +Basic example +************* + +This example estimates 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 diagrams 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:: + + from gudhi.wasserstein.barycenter import lagrangian_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 = 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 ]] |