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-# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-# Author(s): Theo Lacombe
-#
-# Copyright (C) 2019 Inria
-#
-# Modification(s):
-# - YYYY/MM Author: Description of the modification
-
-import numpy as np
-import scipy.spatial.distance as sc
-try:
- import ot
-except ImportError:
- print("POT (Python Optimal Transport) package is not installed. Try to run $ conda install -c conda-forge pot ; or $ pip install POT")
-
-def _proj_on_diag(X):
- '''
- :param X: (n x 2) array encoding the points of a persistent diagram.
- :returns: (n x 2) array encoding the (respective orthogonal) projections of the points onto the diagonal
- '''
- Z = (X[:,0] + X[:,1]) / 2.
- return np.array([Z , Z]).T
-
-
-def _build_dist_matrix(X, Y, order=2., internal_p=2.):
- '''
- :param X: (n x 2) numpy.array encoding the (points of the) first diagram.
- :param Y: (m x 2) numpy.array encoding the second diagram.
- :param internal_p: Ground metric (i.e. norm l_p).
- :param order: exponent for the Wasserstein metric.
- :returns: (n+1) x (m+1) np.array encoding the cost matrix C.
- For 1 <= i <= n, 1 <= j <= m, C[i,j] encodes the distance between X[i] and Y[j], while C[i, m+1] (resp. C[n+1, j]) encodes the distance (to the p) between X[i] (resp Y[j]) and its orthogonal proj onto the diagonal.
- note also that C[n+1, m+1] = 0 (it costs nothing to move from the diagonal to the diagonal).
- '''
- Xdiag = _proj_on_diag(X)
- Ydiag = _proj_on_diag(Y)
- if np.isinf(internal_p):
- C = sc.cdist(X,Y, metric='chebyshev')**order
- Cxd = np.linalg.norm(X - Xdiag, ord=internal_p, axis=1)**order
- Cdy = np.linalg.norm(Y - Ydiag, ord=internal_p, axis=1)**order
- else:
- C = sc.cdist(X,Y, metric='minkowski', p=internal_p)**order
- Cxd = np.linalg.norm(X - Xdiag, ord=internal_p, axis=1)**order
- Cdy = np.linalg.norm(Y - Ydiag, ord=internal_p, axis=1)**order
- Cf = np.hstack((C, Cxd[:,None]))
- Cdy = np.append(Cdy, 0)
-
- Cf = np.vstack((Cf, Cdy[None,:]))
-
- return Cf
-
-
-def _perstot(X, order, internal_p):
- '''
- :param X: (n x 2) numpy.array (points of a given diagram).
- :param internal_p: Ground metric on the (upper-half) plane (i.e. norm l_p in R^2); Default value is 2 (Euclidean norm).
- :param order: exponent for Wasserstein. Default value is 2.
- :returns: float, the total persistence of the diagram (that is, its distance to the empty diagram).
- '''
- Xdiag = _proj_on_diag(X)
- return (np.sum(np.linalg.norm(X - Xdiag, ord=internal_p, axis=1)**order))**(1./order)
-
-
-def wasserstein_distance(X, Y, order=2., internal_p=2.):
- '''
- :param X: (n x 2) numpy.array encoding the (finite points of the) first diagram. Must not contain essential points (i.e. with infinite coordinate).
- :param Y: (m x 2) numpy.array encoding the second diagram.
- :param internal_p: Ground metric on the (upper-half) plane (i.e. norm l_p in R^2); Default value is 2 (euclidean norm).
- :param order: exponent for Wasserstein; Default value is 2.
- :returns: the Wasserstein distance of order q (1 <= q < infinity) between persistence diagrams with respect to the internal_p-norm as ground metric.
- :rtype: float
- '''
- n = len(X)
- m = len(Y)
-
- # handle empty diagrams
- if X.size == 0:
- if Y.size == 0:
- return 0.
- else:
- return _perstot(Y, order, internal_p)
- elif Y.size == 0:
- return _perstot(X, order, internal_p)
-
- M = _build_dist_matrix(X, Y, order=order, internal_p=internal_p)
- a = np.full(n+1, 1. / (n + m) ) # weight vector of the input diagram. Uniform here.
- a[-1] = a[-1] * m # normalized so that we have a probability measure, required by POT
- b = np.full(m+1, 1. / (n + m) ) # weight vector of the input diagram. Uniform here.
- b[-1] = b[-1] * n # so that we have a probability measure, required by POT
-
- # Comptuation of the otcost using the ot.emd2 library.
- # Note: it is the Wasserstein distance to the power q.
- # The default numItermax=100000 is not sufficient for some examples with 5000 points, what is a good value?
- ot_cost = (n+m) * ot.emd2(a, b, M, numItermax=2000000)
-
- return ot_cost ** (1./order)