diff options
Diffstat (limited to 'src/python')
| -rw-r--r-- | src/python/doc/wasserstein_distance_user.rst | 24 | ||||
| -rw-r--r-- | src/python/gudhi/wasserstein.py | 57 | ||||
| -rwxr-xr-x | src/python/test/test_wasserstein_distance.py | 33 |
3 files changed, 89 insertions, 25 deletions
diff --git a/src/python/doc/wasserstein_distance_user.rst b/src/python/doc/wasserstein_distance_user.rst index 94b454e2..9519caa6 100644 --- a/src/python/doc/wasserstein_distance_user.rst +++ b/src/python/doc/wasserstein_distance_user.rst @@ -47,3 +47,27 @@ The output is: .. testoutput:: Wasserstein distance value = 1.45 + +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. +An index of -1 represents the diagonal. + +.. testcode:: + + import gudhi.wasserstein + import numpy as np + + diag1 = np.array([[2.7, 3.7],[9.6, 14.],[34.2, 34.974]]) + diag2 = np.array([[2.8, 4.45], [5, 6], [9.5, 14.1]]) + cost, matching = gudhi.wasserstein.wasserstein_distance(diag1, diag2, matching=True, order=1., internal_p=2.) + + message = "Wasserstein distance value = %.2f, optimal matching: %s" %(cost, matching) + print(message) + +The output is: + +.. testoutput:: + + Wasserstein distance value = 2.15, optimal matching: [[0, 0], [1, 2], [2, -1], [-1, 1]] + diff --git a/src/python/gudhi/wasserstein.py b/src/python/gudhi/wasserstein.py index 13102094..3dd993f9 100644 --- a/src/python/gudhi/wasserstein.py +++ b/src/python/gudhi/wasserstein.py @@ -30,8 +30,10 @@ def _build_dist_matrix(X, Y, order=2., internal_p=2.): :param order: exponent for the Wasserstein metric. :param internal_p: Ground metric (i.e. norm L^p). :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). + For 0 <= i < n, 0 <= j < m, C[i,j] encodes the distance between X[i] and Y[j], + while C[i, m] (resp. C[n, j]) encodes the distance (to the p) between X[i] (resp Y[j]) + and its orthogonal projection onto the diagonal. + note also that C[n, m] = 0 (it costs nothing to move from the diagonal to the diagonal). ''' Xdiag = _proj_on_diag(X) Ydiag = _proj_on_diag(Y) @@ -62,14 +64,20 @@ def _perstot(X, order, internal_p): 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.): +def wasserstein_distance(X, Y, matching=False, 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 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 matching: if True, computes and returns the optimal matching between X and Y, encoded as + a (n x 2) np.array [...[i,j]...], meaning the i-th point in X is matched to + the j-th point in Y, with the convention (-1) represents the diagonal. :param order: exponent for Wasserstein; Default value is 2. - :param internal_p: Ground metric on the (upper-half) plane (i.e. norm L^p in R^2); Default value is 2 (Euclidean norm). - :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 + :param internal_p: Ground metric on the (upper-half) plane (i.e. norm L^p in R^2); + Default value is 2 (Euclidean norm). + :returns: the Wasserstein distance of order q (1 <= q < infinity) between persistence diagrams with + respect to the internal_p-norm as ground metric. + If matching is set to True, also returns the optimal matching between X and Y. ''' n = len(X) m = len(Y) @@ -77,21 +85,40 @@ def wasserstein_distance(X, Y, order=2., internal_p=2.): # handle empty diagrams if X.size == 0: if Y.size == 0: - return 0. + if not matching: + return 0. + else: + return 0., np.array([]) else: - return _perstot(Y, order, internal_p) + if not matching: + return _perstot(Y, order, internal_p) + else: + return _perstot(Y, order, internal_p), np.array([[-1, j] for j in range(m)]) elif Y.size == 0: - return _perstot(X, order, internal_p) + if not matching: + return _perstot(X, order, internal_p) + else: + return _perstot(X, order, internal_p), np.array([[i, -1] for i in range(n)]) 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 + a = np.ones(n+1) # weight vector of the input diagram. Uniform here. + a[-1] = m + b = np.ones(m+1) # weight vector of the input diagram. Uniform here. + b[-1] = n + + if matching: + P = ot.emd(a=a,b=b,M=M, numItermax=2000000) + ot_cost = np.sum(np.multiply(P,M)) + P[-1, -1] = 0 # Remove matching corresponding to the diagonal + match = np.argwhere(P) + # Now we turn to -1 points encoding the diagonal + match[:,0][match[:,0] >= n] = -1 + match[:,1][match[:,1] >= m] = -1 + return ot_cost ** (1./order) , match # 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) + ot_cost = ot.emd2(a, b, M, numItermax=2000000) return ot_cost ** (1./order) diff --git a/src/python/test/test_wasserstein_distance.py b/src/python/test/test_wasserstein_distance.py index 6a6b217b..0d70e11a 100755 --- a/src/python/test/test_wasserstein_distance.py +++ b/src/python/test/test_wasserstein_distance.py @@ -17,7 +17,7 @@ __author__ = "Theo Lacombe" __copyright__ = "Copyright (C) 2019 Inria" __license__ = "MIT" -def _basic_wasserstein(wasserstein_distance, delta, test_infinity=True): +def _basic_wasserstein(wasserstein_distance, delta, test_infinity=True, test_matching=True): diag1 = np.array([[2.7, 3.7], [9.6, 14.0], [34.2, 34.974]]) diag2 = np.array([[2.8, 4.45], [9.5, 14.1]]) diag3 = np.array([[0, 2], [4, 6]]) @@ -51,14 +51,27 @@ def _basic_wasserstein(wasserstein_distance, delta, test_infinity=True): assert wasserstein_distance(diag3, diag4, internal_p=1., order=2.) == approx(np.sqrt(5)) assert wasserstein_distance(diag3, diag4, internal_p=4.5, order=2.) == approx(np.sqrt(5)) - if(not test_infinity): - return + if test_infinity: + diag5 = np.array([[0, 3], [4, np.inf]]) + diag6 = np.array([[7, 8], [4, 6], [3, np.inf]]) - diag5 = np.array([[0, 3], [4, np.inf]]) - diag6 = np.array([[7, 8], [4, 6], [3, np.inf]]) + assert wasserstein_distance(diag4, diag5) == np.inf + assert wasserstein_distance(diag5, diag6, order=1, internal_p=np.inf) == approx(4.) + + + if test_matching: + match = wasserstein_distance(emptydiag, emptydiag, matching=True, internal_p=1., order=2)[1] + assert np.array_equal(match, []) + match = wasserstein_distance(emptydiag, emptydiag, matching=True, internal_p=np.inf, order=2.24)[1] + assert np.array_equal(match, []) + match = wasserstein_distance(emptydiag, diag2, matching=True, internal_p=np.inf, order=2.)[1] + assert np.array_equal(match , [[-1, 0], [-1, 1]]) + match = wasserstein_distance(diag2, emptydiag, matching=True, internal_p=np.inf, order=2.24)[1] + assert np.array_equal(match , [[0, -1], [1, -1]]) + match = wasserstein_distance(diag1, diag2, matching=True, internal_p=2., order=2.)[1] + assert np.array_equal(match, [[0, 0], [1, 1], [2, -1]]) + - assert wasserstein_distance(diag4, diag5) == np.inf - assert wasserstein_distance(diag5, diag6, order=1, internal_p=np.inf) == approx(4.) def hera_wrap(delta): def fun(*kargs,**kwargs): @@ -66,8 +79,8 @@ def hera_wrap(delta): return fun def test_wasserstein_distance_pot(): - _basic_wasserstein(pot, 1e-15, test_infinity=False) + _basic_wasserstein(pot, 1e-15, test_infinity=False, test_matching=True) def test_wasserstein_distance_hera(): - _basic_wasserstein(hera_wrap(1e-12), 1e-12) - _basic_wasserstein(hera_wrap(.1), .1) + _basic_wasserstein(hera_wrap(1e-12), 1e-12, test_matching=False) + _basic_wasserstein(hera_wrap(.1), .1, test_matching=False) |