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Diffstat (limited to 'src/python/doc')
-rw-r--r-- | src/python/doc/wasserstein_distance_user.rst | 72 |
1 files changed, 36 insertions, 36 deletions
diff --git a/src/python/doc/wasserstein_distance_user.rst b/src/python/doc/wasserstein_distance_user.rst index b821b6fa..c24da74d 100644 --- a/src/python/doc/wasserstein_distance_user.rst +++ b/src/python/doc/wasserstein_distance_user.rst @@ -10,10 +10,10 @@ Definition .. include:: wasserstein_distance_sum.inc 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 +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`. +are measured in norm p, for :math:`1 \leq p \leq \infty`. Distance Functions ------------------ @@ -54,9 +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:: @@ -84,7 +84,7 @@ 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 @@ -94,32 +94,32 @@ The output is: 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), +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 +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). +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 -.. figure:: - ./img/barycenter.png - :figclass: align-center - - Illustration of Frechet mean between persistence - diagrams. + Illustration of Frechet mean between persistence + diagrams. .. autofunction:: gudhi.wasserstein.barycenter.lagrangian_barycenter @@ -127,16 +127,16 @@ per diagram). Basic example ************* -This example estimates the Frechet mean (aka Wasserstein barycenter) between +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 +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 +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 +Note that persistence diagrams must be submitted as (n x 2) numpy arrays and must not contain inf values. @@ -152,7 +152,7 @@ Note that persistence diagrams must be submitted as pdiagset = [dg1, dg2, dg3, dg4] bary = lagrangian_barycenter(pdiagset=pdiagset,init=3) - message = "Wasserstein barycenter estimated:" + message = "Wasserstein barycenter estimated:" print(message) print(bary) |