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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")
# 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
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, p=2., q=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 q: Ground metric (i.e. norm l_q).
:param p: 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(q):
C = sc.cdist(X,Y, metric='chebyshev')**p
Cxd = np.linalg.norm(X - Xdiag, ord=q, axis=1)**p
Cdy = np.linalg.norm(Y - Ydiag, ord=q, axis=1)**p
else:
C = sc.cdist(X,Y, metric='minkowski', p=q)**p
Cxd = np.linalg.norm(X - Xdiag, ord=q, axis=1)**p
Cdy = np.linalg.norm(Y - Ydiag, ord=q, axis=1)**p
Cf = np.hstack((C, Cxd[:,None]))
Cdy = np.append(Cdy, 0)
Cf = np.vstack((Cf, Cdy[None,:]))
return Cf
def _perstot(X, p, q):
'''
:param X: (n x 2) numpy.array (points of a given diagram).
:param q: Ground metric on the (upper-half) plane (i.e. norm l_q in R^2); Default value is 2 (Euclidean norm).
:param p: 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=q, axis=1)**p))**(1./p)
def wasserstein_distance(X, Y, p=2., q=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 q: Ground metric on the (upper-half) plane (i.e. norm l_q in R^2); Default value is 2 (euclidean norm).
:param p: exponent for Wasserstein; Default value is 2.
:returns: the p-Wasserstein distance (1 <= p < infinity) with respect to the q-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, p, q)
elif Y.size == 0:
return _perstot(X, p, q)
M = _build_dist_matrix(X, Y, p=p, q=q)
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 squared Wasserstein distance.
# 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./p)
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