def incremental_bary_map_emd(xs, xt, a, b, m1, m2, k):
'''
Compute the incomplete minibatch barycenter mapping
between a source and a target distributions.
(faster for small batch size)
Parameters
----------
- xs : ndarray(ns, d)
source data
- xt : ndarray(nt, d)
target data
- a : ndarray(ns)
source distribution weights
- b : ndarray(nt)
target distribution weights
- m1 : int
source batch size
- m2 : int
target batch size
- k : int
number of batch couples
Returns
-------
- new_xs : ndarray(ns, d)
Transported source measure
- new_xt : ndarray(nt, d)
Transported target measure
'''
new_xs = np.zeros(xs.shape)
new_xt = np.zeros(xt.shape)
Ns = np.shape(xs)[0]
Nt = np.shape(xt)[0]
if m1 < 101:
for i in range(k):
#Test mini batch
sub_xs, sub_weights_a, id_a = small_mini_batch(xs, a, m1, Ns)
sub_xt, sub_weights_b, id_b = small_mini_batch(xt, b, m2, Nt)
sub_M = ot.dist(sub_xs, sub_xt, "sqeuclidean").copy()
G0 = ot.emd(sub_weights_a, sub_weights_b, sub_M)
new_xs[id_a] += G0.dot(xt[id_b])
new_xt[id_b] += G0.T.dot(xs[id_a])
else:
for i in range(k):
#Test mini batch
sub_xs, sub_weights_a, id_a = mini_batch(xs, a, m1, Ns)
sub_xt, sub_weights_b, id_b = mini_batch(xt, b, m2, Nt)
sub_M = ot.dist(sub_xs, sub_xt, "sqeuclidean").copy()
G0 = ot.emd(sub_weights_a, sub_weights_b, sub_M)
new_xs[id_a] += G0.dot(xt[id_b])
new_xt[id_b] += G0.T.dot(xs[id_a])
return 1. / k * Ns * new_xs, 1. / k * Nt * new_xt
def _compute_copula_ot_dependence(empirical: np.array, target: np.array,
forget: np.array, n_obs: int) -> float:
"""
Calculates optimal copula transport dependence measure.
:param empirical: (np.array) Empirical copula.
:param target: (np.array) Target copula.
:param forget: (np.array) Forget copula.
:param nb_obs: (int) Number of observations.
:return: (float) Optimal copula transport dependence.
"""
# Uniform distribution on samples
t_measure, f_measure, e_measure = (np.ones(
(n_obs, )) / n_obs, np.ones((n_obs, )) / n_obs, np.ones(
(n_obs, )) / n_obs)
# Compute the ground distance matrix between locations
gdist_e2t = ot.dist(empirical, target)
gdist_e2f = ot.dist(empirical, forget)
# Compute the optimal transport matrix
e2t_ot = ot.emd(t_measure, e_measure, gdist_e2t)
e2f_ot = ot.emd(f_measure, e_measure, gdist_e2f)
# Compute the optimal transport distance:
# <optimal transport matrix, ground distance matrix>_F
e2t_dist = np.trace(np.dot(np.transpose(e2t_ot), gdist_e2t))
e2f_dist = np.trace(np.dot(np.transpose(e2f_ot), gdist_e2f))
# Compute the copula ot dependence measure
ot_measure = 1 - e2t_dist / (e2f_dist + e2t_dist)
return ot_measure
def test_emd_emd2_devices_tf():
if not tf:
return
nx = ot.backend.TensorflowBackend()
n_samples = 100
n_features = 2
rng = np.random.RandomState(0)
x = rng.randn(n_samples, n_features)
y = rng.randn(n_samples, n_features)
a = ot.utils.unif(n_samples)
M = ot.dist(x, y)
# Check that everything stays on the CPU
with tf.device("/CPU:0"):
ab, Mb = nx.from_numpy(a, M)
Gb = ot.emd(ab, ab, Mb)
w = ot.emd2(ab, ab, Mb)
nx.assert_same_dtype_device(Mb, Gb)
nx.assert_same_dtype_device(Mb, w)
if len(tf.config.list_physical_devices('GPU')) > 0:
# Check that everything happens on the GPU
ab, Mb = nx.from_numpy(a, M)
Gb = ot.emd(ab, ab, Mb)
w = ot.emd2(ab, ab, Mb)
nx.assert_same_dtype_device(Mb, Gb)
nx.assert_same_dtype_device(Mb, w)
assert nx.dtype_device(Gb)[1].startswith("GPU")
def get_lang_mapping(self, lang1, lang2, metric, entreg):
path = self._get_shortest_path_from_lang1_to_lang2(lang1, lang2)
print(path)
mapping = None
for i in range(len(path)):
a = self.lang_dict[path[i][0]]
a = self.project_into_lang_space(a, self.args.lang_space)
b = self.lang_dict[path[i][0]].child[path[i][1]]
b = self.project_into_lang_space(b, self.args.lang_space)
if path[i][2]:
plan = ot.emd(b.freq, a.freq, build_MXY(b.projected_matrix, a.projected_matrix))
else:
plan = ot.emd(a.freq, b.freq, build_MXY(a.projected_matrix, b.projected_matrix))
if mapping is None:
mapping = plan
else:
mapping = np.matmul(mapping, plan)
a.projected_matrix = None
b.projected_matrix = None
return mapping
def test_warnings():
n = 100 # nb bins
m = 100 # nb bins
mean1 = 30
mean2 = 50
# bin positions
x = np.arange(n, dtype=np.float64)
y = np.arange(m, dtype=np.float64)
# Gaussian distributions
a = gauss(n, m=mean1, s=5) # m= mean, s= std
b = gauss(m, m=mean2, s=10)
# loss matrix
M = ot.dist(x.reshape((-1, 1)), y.reshape((-1, 1)))**(1. / 2)
print('Computing {} EMD '.format(1))
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
print('Computing {} EMD '.format(1))
ot.emd(a, b, M, numItermax=1)
assert "numItermax" in str(w[-1].message)
def forward_seq(self, x_train, x_test):
N = self.Nmasses
(Pl_train, P_train) = particleApproximation_v0(x_train, N)
(Pl_test, P_test) = particleApproximation_v0(x_test, N)
Pl_tem = 0
for a in range(2): #x_train.shape[0]):
t = Pl_train[a]
Pl_tem = Pl_tem + t
Pl_tem = Pl_tem / 2 #x_train.shape[0]
P_tem = np.ones((N, )) / float(N)
#Pl_tem_vec=np.reshape(Pl_tem,(Pl_tem.shape[0]*Pl_tem.shape[1],),order='F')
V = list()
M = x_train.shape[0]
for ind in range(M):
Ni = Pl_train[ind].shape[0]
C = ot.dist(Pl_train[ind], Pl_tem)
b = P_tem # b=np.ones((N,))/float(N)
a = P_train[ind] # a=np.ones((Ni,))/float(Ni)
p = ot.emd(a, b, C) # exact linear program
#V.append(np.matmul((N*p).T,Pl_train[ind])-Pl_tem)
V.append(np.matmul((N * p).T, Pl_train[ind]) +
Pl_tem) # already giving transport displacement?
V = np.asarray(V)
x_train_hat = np.zeros((len(V), V[0].shape[0] * V[0].shape[1]))
for a in range(len(V)):
x_train_hat[a, :] = np.reshape(V[a],
(V[0].shape[0] * V[0].shape[1], ),
order='F')
V = list()
M = x_test.shape[0]
for ind in range(M):
Ni = Pl_test[ind].shape[0]
C = ot.dist(Pl_test[ind], Pl_tem)
b = P_tem # b=np.ones((N,))/float(N)
a = P_test[ind] # a=np.ones((Ni,))/float(Ni)
p = ot.emd(a, b, C) # exact linear program
#V.append(np.matmul((N*p).T,Pl_test[ind])-Pl_tem)
V.append(np.matmul((N * p).T, Pl_test[ind]) + Pl_tem)
V = np.asarray(V)
x_test_hat = np.zeros((len(V), V[0].shape[0] * V[0].shape[1]))
for a in range(len(V)):
x_test_hat[a, :] = np.reshape(V[a],
(V[0].shape[0] * V[0].shape[1], ),
order='F')
return x_train_hat, x_test_hat, Pl_tem, P_tem
def computeTransportLaplacianSymmetric_fw(distances, Ss, St, xs, xt, reg=1e-9, regls=0, reglt=0, solver=None,
nbitermax=400, thr_stop=1e-8, step='opt', **kwargs):
distribS = np.ones((xs.shape[0],)) / xs.shape[0]
distribT = np.ones((xt.shape[0],)) / xt.shape[0]
Ls = get_laplacian(Ss)
Lt = get_laplacian(St)
loop = True
transp = ot.emd(distribS, distribT, distances)
niter = 0
while loop:
old_transp = transp.copy()
G = np.asarray(regls * get_gradient1(Ls, xt, old_transp) + reglt * get_gradient2(Lt, xs, old_transp))
transp0 = ot.emd(distribS, distribT, distances + G)
E = transp0 - old_transp
# Ge=get_gradient(E,K)
if step == 'opt':
# optimal step size !!!
tau = max(0, min(1, (-np.sum(E * distances) - np.sum(E * G)) / (
2 * regls * quadloss1(E, Ls, xt) + 2 * reglt * quadloss2(E, Lt, xs))))
else:
# other step size just in case
tau = 2. / (niter + 2) # print "tau:",tau
transp = old_transp + tau * E
# print "loss:",np.sum(transp*distances)+quadloss(transp,K)/2
if niter >= nbitermax:
loop = False
err = np.sum(np.abs(transp - old_transp))
if err < thr_stop:
loop = False
# print niter
niter += 1
if niter % 1000 == 0:
print('{:5s}|{:12s}'.format('It.', 'Err') + '\n' + '-' * 19)
print('{:5d}|{:8e}|'.format(niter, err))
# print "loss:",np.sum(transp*distances)+quadloss(transp,K)/2
return transp
def sinkhorn_mapping(set_1, set_2):
"""http://pot.readthedocs.io/en/stable/auto_examples/plot_OT_2D_samples.html"""
a, b = np.ones((len(set_1), )) / len(set_1), np.ones(
(len(set_2), )) / len(set_2)
arr_1 = _generate_arr(set_1)
arr_2 = _generate_arr(set_2)
switch_to_cartesian(arr_1, 80 * 400)
switch_to_cartesian(arr_2, 80 * 400)
M = ot.dist(arr_1, arr_2)
G0 = ot.emd(a, b, M)
counter = 0
for i in G0:
inner_counter = 0
for j in i:
if j > 0.003:
print(arr_1[counter], arr_2[inner_counter], j, sep=", ")
inner_counter += 1
counter += 1
pl.figure(4)
for i in range(arr_1.shape[0]):
for j in range(arr_2.shape[0]):
if G0[i, j] > 0.003:
pl.plot([arr_1[i, 0], arr_2[j, 0]], [arr_1[i, 1], arr_2[j, 1]])
pl.plot(arr_1[:, 0], arr_1[:, 1], '+b', label='Source samples')
pl.plot(arr_2[:, 0], arr_2[:, 1], 'xr', label='Target samples')
pl.show()
def test(source_samples,
target_samples,
weight_function):
"""
:param source_samples: array (n_source, feature)
:param target_samples: array (n_target, feature)
:param weight_function: function determine distance between two samples
:return:
"""
assert source_samples.shape[1] == target_samples.shape[1]
# Employ uniform distribution over all data as empirical distribution (not a histogram)
source_dist = np.ones((len(source_samples), )) / len(source_samples)
target_dist = np.ones((len(target_samples), )) / len(target_samples)
# print('source:', source_dist.shape, np.sum(source_dist))
# print('target:', target_dist.shape, np.sum(target_dist))
# build cost matrix (n_source, n_target)
cost_matrix = np.array([[float(weight_function(__i, __o)) for __i in target_samples] for __o in source_samples])
print('cost :\n', cost_matrix, cost_matrix.shape)
# derive optimal transport based on network simplex algorithm
# Bonneel, N., Van De Panne, M., Paris, S., & Heidrich, W. (2011, December).
# Displacement interpolation using Lagrangian mass transport.
# In ACM Transactions on Graphics (TOG) (Vol. 30, No. 6, p. 158). ACM.
optimal_transport = ot.emd(a=source_dist, b=target_dist, M=cost_matrix)
return optimal_transport
def emd(xs, xt, metric='euclidean', numItermax=2**17, **kwargs):
if len(xs.shape) == 1:
xs = xs.reshape(-1, 1)
if len(xt.shape) == 1:
xt = xt.reshape(-1, 1)
M = cdist(xs, xt, metric)
if (M == 0).all():
return 0.
M2 = M**2
a = np.ones(len(xs)) / len(xs)
b = np.ones(len(xt)) / len(xt)
for i in range(3):
try:
F = ot.emd(a, b, M2, numItermax, **kwargs)
# np.isclose(np.sum(F), 1)
return np.sum(M * F)
except UserWarning:
numItermax = 2 * numItermax
assert False, "No conversion reached. Try to increase numItermax!"
def compute_ot_loss_matrix(y: np.ndarray,
y_hat: np.ndarray,
D: np.ndarray,
ot_niters=10**5):
"""
Solve the optimal transport problem for the image pixels, and return the OT
permutation matrix Pi.
:param y: the ground-truth image.
:param y_hat: the predicted image.
:param D: the distance matrix; generate via make_distance_matrix(y.shape[0])
:param y_hat_as_logits: if True, y_hat is provided as logits.
:return: Pi, the optimal transport matrix, of shape [d**2, d**2]. The (i,j) entry
in Pi represents the cost of moving pixel i in y_hat to pixel j in y.
"""
assert_array_finite(y)
assert_array_finite(y_hat)
assert_array_nonnegative(y)
assert_array_nonnegative(y_hat)
np.testing.assert_array_equal(y.shape[0],
y.shape[1]) # check images are square
np.testing.assert_array_equal(y.shape,
y_hat.shape) # check images same size
y_hist = normalize_to_histogram(y)
y_hat_hist = normalize_to_histogram(y_hat)
PI = ot.emd(y_hat_hist, y_hist, D, numItermax=ot_niters)
return PI
def diagonality(ir_dft, ir_dftb):
#normalize spectra
ir_dft = [i / np.sum(i) for i in ir_dft]
ir_dftb = [i / np.sum(i) for i in ir_dftb]
#diagonality of P https://math.stackexchange.com/questions/1392491/measure-of-how-much-diagonal-a-matrix-is
Y, X = np.meshgrid(np.linspace(0, 1, ir_dft[0].size),
np.linspace(0, 1, ir_dft[0].size))
C = abs(Y - X)**2
def dist(P):
j = np.ones(P.shape[0])
r = np.arange(P.shape[0])
r2 = r**2
n = j @ P @ j.T
sum_x = r @ P @ j.T
sum_y = j @ P @ r.T
sum_x2 = r2 @ P @ j.T
sum_y2 = j @ P @ r2.T
sum_xy = r @ P @ r.T
return (n * sum_xy - sum_x * sum_y) / (np.sqrt(n * sum_x2 - sum_x**2) *
np.sqrt(n * sum_y2 - sum_y**2))
# print('Case (Diagonality)')
d = np.zeros((len(ir_dft), len(ir_dftb)))
for i, a in enumerate(ir_dft):
for j, b in enumerate(ir_dftb):
# P = sink.sinkhorn(a,b, 0.003).P
P = ot.emd(a, b, C)
d[i, j] = dist(P)
return d
def match_spots_using_spatial_heuristic(X,
Y,
use_ot: bool = True) -> np.ndarray:
"""
Calculates and returns a mapping of spots using a spatial heuristic.
Args:
X (array-like, optional): Coordinates for spots X.
Y (array-like, optional): Coordinates for spots Y.
use_ot: If ``True``, use optimal transport ``ot.emd()`` to calculate mapping. Otherwise, use Scipy's ``min_weight_full_bipartite_matching()`` algorithm.
Returns:
Mapping of spots using a spatial heuristic.
"""
n1, n2 = len(X), len(Y)
X, Y = norm_and_center_coordinates(X), norm_and_center_coordinates(Y)
dist = scipy.spatial.distance_matrix(X, Y)
if use_ot:
pi = ot.emd(np.ones(n1) / n1, np.ones(n2) / n2, dist)
else:
row_ind, col_ind = scipy.sparse.csgraph.min_weight_full_bipartite_matching(
scipy.sparse.csr_matrix(dist))
pi = np.zeros((n1, n2))
pi[row_ind, col_ind] = 1 / max(n1, n2)
if n1 < n2:
pi[:, [(j not in col_ind) for j in range(n2)]] = 1 / (n1 * n2)
elif n2 < n1:
pi[[(i not in row_ind) for i in range(n1)], :] = 1 / (n1 * n2)
return pi
def OT_emd(Xl, Xr):
# loss matrix
C = ot.dist(Xl, Xr)
M = C / C.max()
n = len(Xl)
m = len(Xr)
print(n, m)
a, b = np.ones((n, )) / n, np.ones((m, )) / m
G0 = ot.emd(a, b, M)
plt.figure(3)
plt.imshow(G0, interpolation='nearest')
plt.title('OT matrix G0')
plt.figure(4)
ot.plot.plot2D_samples_mat(Xl[:, :2], Xr[:, :2], G0, c=[.5, .5, 1])
plt.plot(Xl[:, 0], Xl[:, 1], '+b', label='Source samples')
plt.plot(Xr[:, 0], Xr[:, 1], 'xr', label='Target samples')
plt.legend(loc=0)
plt.title('OT matrix with samples')
return (G0)
def OT_scores_emd(Xl, Xr, scoresl, scoresr, mu=0.5):
# loss matrix
print(scoresl)
C = ot.dist(Xl, Xr) + mu * ot.dist(np.expand_dims(scoresl, axis=1),
np.expand_dims(scoresr, axis=1))
M = C / C.max()
n = len(Xl)
m = len(Xr)
print(n, m)
a, b = scoresl, scoresr
G0 = ot.emd(a, b, M)
plt.figure(3)
plt.imshow(G0, interpolation='nearest')
plt.title('OT matrix G0')
plt.figure(4)
ot.plot.plot2D_samples_mat(Xl[:, :2], Xr[:, :2], G0, c=[.5, .5, 1])
plt.plot(Xl[:, 0], Xl[:, 1], '+b', label='Source samples')
plt.plot(Xr[:, 0], Xr[:, 1], 'xr', label='Target samples')
plt.legend(loc=0)
plt.title('OT matrix with samples')
return (G0)
def get_node_cell_type_intersection(tree):
# find the amount of intersection each node has with each cell type
labels, suppl_image_coords = get_image_coords(supplementary_data)
layers = list(get_layers(tree))
cost_matrices, costs = [], []
for layer_ind, layer in enumerate(layers):
if len(layer) == 0:
continue
print('Layer', layer_ind, 'number of nodes', len(layer))
node_prop = np.array([node.coords.shape[0] for node in layer])
node_prop = node_prop / node_prop.sum()
suppl_prop = np.array(
[image_coords.shape[0] for image_coords in suppl_image_coords])
suppl_prop = suppl_prop / suppl_prop.sum()
cost_matrix = get_intersection(layer, suppl_image_coords)
transport_matrix, log = ot.emd(node_prop,
suppl_prop,
1 - cost_matrix,
log=True)
costs.append(log['cost'])
cost_matrices.append(cost_matrix)
cost_min_ind = np.argmin(costs)
cost_matrix = cost_matrices[cost_min_ind]
layer = layers[cost_min_ind]
print('Optimal layer is {} with {} clusters.'.format(
cost_min_ind, len(layer)))
annotate_optimal_layer(labels, cost_matrix, layer)
def weighted_barycenter_algorithm(ls, test_func, X_orig, Yi_orig, bi, lambdas, tol=1e-8, metric='euclidean', reg=1e-2, maxiter=20, bregmanmaxiter=30):
"""
k : number of supports in X
X_orig : init of barycenter (k * d)
Yi_orig : list of distributions size (k_i * d)
bi : list of weights size (k_i)
tol: tolerance
"""
assert(len(Yi_orig) == len(bi))
assert(len(X_orig[0]) == len(Yi_orig[0][0]))
X = X_orig
Yi = Yi_orig
displacement = 1
niter = 0
while (displacement > tol and niter < maxiter):
X_prev = X
a = compute_barycenter_weight(X, Yi, bi, lambdas, tol=tol, maxiter=bregmanmaxiter, reg=reg)
Tsum = np.zeros(X.shape)
for i in range(0, len(bi)):
M = build_MXY(X, Yi[i], metric=metric)
#T = ot.sinkhorn(a, bi[i], M, reg)
T = ot.emd(a, bi[i], M)
Tsum = Tsum + lambdas[i] * np.reshape(1. / a, (-1, 1)) * np.matmul(T, Yi[i])
displacement = np.sum(np.square(Tsum - X))
print("~~~~epoch "+str(niter)+"~~~~")
#i = ls.index('en')
#for j in range(len(ls)):
# if i!=j and (not ls[i].isdigit()) and (not ls[j].isdigit()):
# mapping = ot.emd(bi[i], a, build_MXY(Yi[i], X))
# mapping2 = ot.emd(a, bi[j], build_MXY(X, Yi[j]))
# print("="*20+"begin testing mapping for "+ls[i]+" and "+ls[j]+"="*21)
# test_func(ls[i], ls[j], np.dot(mapping, mapping2))
# mapping = None
# mapping2 = None
#for i in range(len(ls)):
# for j in range(len(ls)):
# if i!=j and (not ls[i].isdigit()) and (not ls[j].isdigit()):
# mapping = ot.emd(bi[i], a, build_MXY(Yi[i], X))
# mapping2 = ot.emd(a, bi[j], build_MXY(X, Yi[j]))
# print("="*20+"begin testing mapping for "+ls[i]+" and "+ls[j]+"="*21)
# try:
# test_func(ls[i], ls[j], np.dot(mapping, mapping2))
# except:
# print("failed to eval on "+ls[i]+" and "+ls[j])
# mapping = None
# mapping2 = None
X = Tsum
niter += 1
return X, a
def rbd_wasserstein_approx2(f1,f2):
im1 = rbd_read(f1).flatten().reshape(-1,1)
im2 = rbd_read(f2).flatten().reshape(-1,1)
gauss1 = mix.GaussianMixture(2).set_params(tol=1e-1).fit(im1)
gauss2 = mix.GaussianMixture(2).set_params(tol=1e-1).fit(im2)
m11,m12 = gauss1.means_.flatten()
v11,v12 = gauss1.covariances_.flatten()
p11,p12 = gauss1.weights_.flatten()
m21,m22 = gauss2.means_.flatten()
v21,v22 = gauss2.covariances_.flatten()
p21,p22 = gauss2.weights_.flatten()
d1 = np.array([p11,p12,0,0])
d2 = np.array([0,0,p21,p22])
m = np.array([m11,m12,m21,m22])
v = np.array([v11,v12,v21,v22])
weight_matrix = np.zeros((len(d1),len(d2)))
for i in range(len(d1)):
for j in range(len(d1)):
weight_matrix[i,j] = gauss_wasserstein(m[i],v[i],m[j],v[j])
return(ot.emd(d1,d2,weight_matrix))
def solve_optimal_transport_problem(self, ):
self.n = len(self.weights_model1)
self.a, self.b = np.ones((self.n,)) / self.n, np.ones((self.n,)) / self.n # uniform distribution on samples
# loss matrix
self.M = ot.dist(self.weights_model1, self.weights_model2)
self.M /= self.M.max()
self.G0 = ot.emd(self.a, self.b, self.M)
def compute_gamma(self, pred):
'''
Function to compute the OT between the target and source samples.
:return:Gamma the OT matrix
'''
# Reshaping the samples into vectors of dimensions number of modalities * patch_dimension.
# train_vecs are of shape (batch_size, d)
train_vec_source = np.reshape(self.image_representation_source, (self.batch_size, self.image_representation_source.shape[1]*
self.image_representation_source.shape[2]*
self.image_representation_source.shape[3]*
self.image_representation_source.shape[4]))
train_vec_target = np.reshape(self.image_representation_target, (self.batch_size, self.image_representation_target.shape[1]*
self.image_representation_target.shape[2]*
self.image_representation_target.shape[3]*
self.image_representation_target.shape[4]))
# Same for the ground truth but the GT is the same for both modalities
truth_vec_source = np.reshape(self.train_batch[1][:self.batch_size],
(self.batch_size, self.config.patch_shape[0]*self.config.patch_shape[1]*self.config.patch_shape[2]))
pred_vec_source = np.reshape(pred[:self.batch_size],
(self.batch_size, self.config.patch_shape[0]*self.config.patch_shape[1]*self.config.patch_shape[2]))
# We don't have information on target labels
pred_vec_target = np.reshape(pred[self.batch_size:],
(self.batch_size, self.config.patch_shape[0]*self.config.patch_shape[1]*self.config.patch_shape[2]))
# Compute the distance between samples and between the source_truth and the target prediction.
C0 = cdist(train_vec_source, train_vec_target, metric="sqeuclidean")
C1 = cdist(truth_vec_source, pred_vec_target, metric=self.config.jdot_distance)
C = K.get_value(self.jdot_alpha)*C0+K.get_value(self.jdot_beta)*C1
# Computing gamma using the OT library
gamma = ot.emd(ot.unif(self.batch_size), ot.unif(self.batch_size), C)
return gamma
def gromov_wasserstein_distance_latent_space_rand_emd(data_path, num_labels,
num_clusters,
result_path, args):
import scipy as sp
import matplotlib.pylab as pl
import ot
# z = np.load(data_path+ "/L-1/z.npy") # -1 means no discrimation for labelsa, the same vae transform , orthogonal concept to whether cluster on this z space or use other mehtod to split into clusters
z = np.load(
data_path + "/L-1" + config.z_name, allow_pickle=True
) # -1 means no discrimation for labelsa, the same vae transform , orthogonal concept to whether cluster on this z space or use other mehtod to split into clusters
np.random.shuffle(z)
results = {}
mat = np.zeros((num_clusters, num_clusters))
from sklearn.model_selection import KFold
kf = KFold(n_splits=num_clusters)
i = 0
cluster_idx = {}
for train_eval_idx, test_idx in kf.split(z):
cluster_idx[str(i)] = test_idx
i = i + 1
i = 0
print(z.shape)
for i in range(num_clusters):
xs = z[cluster_idx[str(i)]]
print(xs.shape)
for j in range(num_clusters):
xt = z[cluster_idx[str(j)]]
print(xt.shape)
# Compute distance kernels, normalize them and then display
n_samples = min(xs.shape[0], xt.shape[0])
if args.debug == True:
n_samples = 100
xs = xs[:n_samples]
xt = xt[:n_samples]
M = sp.spatial.distance.cdist(xt, xs)
M /= M.max()
ds, dt = np.ones((len(xs), )) / len(xs), np.ones(
(len(xt), )) / len(xt)
g0, loss = ot.emd(ds, dt, M, log=True)
print(
'Gromov-Wasserstein distances between {}_{} clusters: {}--{} '.
format(i, j, str(loss), str(np.sum(g0))))
#results[str(i)+str(j)]={"GW":log0['gw_dist'],"EGW":log['gw_dist']}
results[str(i) + str(j)] = loss["cost"]
mat[i, j] = loss["cost"]
#pl.figure(1, (10, 5))
#pl.subplot(1, 2, 1)
#pl.imshow(gw0, cmap='jet')
#pl.title('Gromov Wasserstein')
#pl.subplot(1, 2, 2)
#pl.imshow(gw, cmap='jet')
#pl.title('Entropic Gromov Wasserstein')
#pl.savefig(result_path + "/WD_TSNE{}_{}.jpg".format(i,j))
# print(results)
print(mat)
with open("wd_rand.txt", 'a') as lf:
lf.write(str(results))
return results
def test_emd_1d_emd2_1d_with_weights():
# test emd1d gives similar results as emd
n = 20
m = 30
rng = np.random.RandomState(0)
u = rng.randn(n, 1)
v = rng.randn(m, 1)
w_u = rng.uniform(0., 1., n)
w_u = w_u / w_u.sum()
w_v = rng.uniform(0., 1., m)
w_v = w_v / w_v.sum()
M = ot.dist(u, v, metric='sqeuclidean')
G, log = ot.emd(w_u, w_v, M, log=True)
wass = log["cost"]
G_1d, log = ot.emd_1d(u, v, w_u, w_v, metric='sqeuclidean', log=True)
wass1d = log["cost"]
wass1d_emd2 = ot.emd2_1d(u, v, w_u, w_v, metric='sqeuclidean', log=False)
wass1d_euc = ot.emd2_1d(u, v, w_u, w_v, metric='euclidean', log=False)
# check loss is similar
np.testing.assert_allclose(wass, wass1d)
np.testing.assert_allclose(wass, wass1d_emd2)
# check loss is similar to scipy's implementation for Euclidean metric
wass_sp = wasserstein_distance(u.reshape((-1, )), v.reshape((-1, )), w_u,
w_v)
np.testing.assert_allclose(wass_sp, wass1d_euc)
# check constraints
np.testing.assert_allclose(w_u, G.sum(1))
np.testing.assert_allclose(w_v, G.sum(0))
def wass1dim(data1, data2, numBins=200):
''' Compare two one-dimensional arrays by the
Wasserstein metric (https://en.wikipedia.org/wiki/Wasserstein_metric).
The input data should have outliers removed.
Parameters
----------
data1, data2: two one-dimensional arrays to compare.
numBins: the number of bins.
Outputs
-------
result: the computed Wasserstein metric.
'''
numBins = 200 ## number of bins
upper = np.max((data1.max(), data2.max()))
lower = np.min((data1.min(), data2.min()))
xbins = np.linspace(lower, upper, numBins + 1)
density1, _ = np.histogram(data1, density=False, bins=xbins)
density2, _ = np.histogram(data2, density=False, bins=xbins)
density1 = density1 / np.sum(density1)
density2 = density2 / np.sum(density2)
# pairwise distance matrix between bins
distMat = distance_matrix(xbins[1:].reshape(numBins, 1),
xbins[1:].reshape(numBins, 1))
M = distMat
T = ot.emd(density1, density2, M) # optimal transport matrix
result = np.sum(T * M) # the objective data
return result
def compute_deepjdot_loss(features_source, ys_pred, ys, features_target,
yt_pred, gamma_criterion, g_criterion):
# Compute the euclidian distance in the feature space
#C0 = cdist(features_source.detach().cpu().numpy(),
# features_target.detach().cpu().numpy(), p=0.2)
C0 = torch.square(torch.cdist(features_source, features_target, p=2.0))
# Compute the loss function of labels
#C1 = F.cross_entropy(yt_pred, ys)
classes = torch.arange(yt_pred.shape[1]).reshape(1, yt_pred.shape[1])
one_hot_ys = (ys.unsqueeze(1) == classes.to(device=c.device)).float()
C1 = torch.square(
torch.cdist(one_hot_ys, F.softmax(yt_pred, dim=1), p=2.0))
C = c.alpha * C0 + c.tloss * C1
# Compute the gamma function
#gamma = ot.emd(ot.unif(features_source.shape[0]),
# ot.unif(features_target.shape[0]), C)
gamma = ot.emd(
torch.from_numpy(ot.unif(
features_source.shape[0])).to(device=c.device),
torch.from_numpy(ot.unif(
features_target.shape[0])).to(device=c.device), C)
# ot.emd: solve the OT problem for the pdfs of both source and target features
# ot.unif: return an histogram of the arguments
# Align Loss
gamma_loss = gamma_criterion(features_source, features_target, gamma)
# gamma loss get the fetures of the source, the features of the target
# and gamma. It first performs the L2 distance between the features and
# then return self.jdot_alpha * dnn.K.sum(self.gamma * (gdist))
# Classifier Loss
clf_loss = g_criterion(ys, ys_pred, yt_pred, gamma)
return clf_loss, gamma_loss, clf_loss + gamma_loss
def _match_shorter(self, shorter):
"""compute mapping if new points are less than known points"""
M, known, _ = _dist_closest(self.points, shorter)
G = ot.emd([], [], M[known])
result = np.empty((len(shorter, )), dtype=int)
result[np.argmax(G, axis=1)] = known
return result
def persistence_wasserstein_distance(x: np.ndarray, y: np.ndarray,
ground_distance: np.ndarray) -> float:
"""Compute an approximation of Persistence Wasserstein_1 distance
between persistenced iagrams with vector representations ``x`` and ``y``
using the ground distance provided.
Parameters
----------
x: array of shape (n_gaussians,)
The vectorization of the first persistence diagram
y: array of shape (n_gaussians,)
The vectorization of the first persistence diagram
ground_distance: array of shape (n_gaussians + 1, n_gaussians + 1)
The amended ground-distance as output by ``add_birth_death_line``
Returns
-------
dist: float
Ann approximation of Persistence Wasserstein_1 distance
between persistenced iagrams with vector representations
``x`` and ``y``
"""
x_a = np.append(x, y.sum())
x_a /= x_a.sum()
y_a = np.append(y, x.sum())
y_a /= y_a.sum()
plan = ot.emd(x_a, y_a, ground_distance)
return (x.sum() + y.sum()) * (plan * ground_distance).sum()
def graph_d(self, graph1, graph2):
""" Compute the Wasserstein distance between two graphs. Uniform weights are used.
Parameters
----------
graph1 : a Graph object
graph2 : a Graph object
Returns
-------
The Wasserstein distance between the features of graph1 and graph2
"""
nodes1 = graph1.nodes()
nodes2 = graph2.nodes()
t1masses = np.ones(len(nodes1)) / len(nodes1)
t2masses = np.ones(len(nodes2)) / len(nodes2)
x1 = self.reshaper(graph1.all_matrix_attr())
x2 = self.reshaper(graph2.all_matrix_attr())
if self.features_metric == 'dirac':
f = lambda x, y: x != y
M = ot.dist(x1, x2, metric=f)
else:
M = ot.dist(x1, x2, metric=self.features_metric)
if np.max(M) != 0:
M = M / np.max(M)
self.M = M
transp = ot.emd(t1masses, t2masses, M)
self.transp = transp
return np.sum(transp * M)
def pot_wasserstein_mapper(net1, net2, metric_space, p=None, q=None):
""" Computes vanilla EMD (over Hausdorff dist) for mapper graphs
Parameters
----------
net1 : lightweight_mapper.Network
Mapper graph
net2 : lightweight_mapper.Network
Mapper graph
metric_space : np.array
Pairwise distance matrix
p : np.array - nx1
Distribution over nodes corresponding to net1
q : np.array - nx1
Distribution over nodes corresponding to net2
Returns
-------
EMD (Cost = Hausdorff dist)
"""
C3 = network_merge_distance(net1, net2, metric_space)
if p is None or q is None:
p = np.diag(net1.adjacency_matrix.toarray())
p = p / p.sum()
q = np.diag(net2.adjacency_matrix.toarray())
q = q / q.sum()
gw_dist = ot.emd2(p, q, C3)
params = ot.emd(p, q, C3)
return gw_dist, params
def makeTransportPlan(self):
if self.source_data and self.target_data:
if self.source_data_size == self.target_data_size:
loss_matrix = ot.dist(self.source_data, self.target_data)
loss_matrix = loss_matrix / loss_matrix.max()
if not self.source_weight:
self.source_weight = np.ones(
(self.source_data_size, )) / self.source_data_size
print(
"The Source weights are intiialized to one. If custome weight please load weight."
)
if not self.target_weight:
self.target_weight = np.ones(
(self.target_data_size, )) / self.target_data_size
print(
"The Target weights are intiialized to one. If custome weight please load weight."
)
transport = ot.emd(self.source_weight,
self.target_weight,
loss_matrix,
log=True)
print("Transport Plan Complete")
print("The cost is: {}".format(transport[1]['cost']))
self.transport_plan = transport[0]
return transport
else:
print(
"Optimal Transport Plan not complete due to mismatch in Source and Target Size."
)
return
else:
print(
"Optimal Transport Plan not complete. Please add Source & Target data and rerun."
)
return
def gnpr_distance(x: np.array,
y: np.array,
theta: float,
n_bins: int = 50) -> float:
"""
Calculates the empirical distance between two random variables under the Generic Non-Parametric Representation
(GNPR) approach.
Formula for the distance is taken from https://www.researchgate.net/publication/322714557 (p.72).
Parameter theta defines what type of information dependency is being tested:
- for theta = 0 the distribution information is tested
- for theta = 1 the dependence information is tested
- for theta = 0.5 a mix of both information types is tested
With theta in [0, 1] the distance lies in the range [0, 1] and is a metric.
(See original work for proof, p.71)
This method is modified as it uses 1D Optimal Transport Distance to measure
distribution distance. This solves the issue of defining support and choosing
a number of bins. The number of bins can be given as an input to speed up calculations.
Big numbers of bins can take a long time to calculate.
:param x: (np.array/pd.Series) X vector.
:param y: (np.array/pd.Series) Y vector (same number of observations as X).
:param theta: (float) Type of information being tested. Falls in range [0, 1].
:param n_bins: (int) Number of bins to use to split the X and Y vector observations.
(100 by default)
:return: (float) Distance under GNPR approach.
"""
# Number of observations
num_obs = x.shape[0]
# Calculating the d_1 distance
dist_1 = 3 / (num_obs * (num_obs**2 - 1)) * (np.power(x - y, 2).sum())
# Binning observations
x_binned = pd.Series(np.histogram(x, bins=n_bins)[0]) / num_obs
y_binned = pd.Series(np.histogram(y, bins=n_bins)[0]) / num_obs
# Bin positions
bins = np.linspace(0, 1, n_bins)
# Loss matrix
loss_matrix = ot.dist(bins.reshape((n_bins, 1)), bins.reshape((n_bins, 1)))
loss_matrix /= loss_matrix.max()
# Optimal transportation matrix
ot_matrix = ot.emd(x_binned.sort_values(), y_binned.sort_values(),
loss_matrix)
# Optimal transport distance
dist_0 = np.trace(np.dot(np.transpose(ot_matrix), loss_matrix))
# Calculating the GNPR distance
distance = theta * dist_1 + (1 - theta) * dist_0
return distance**(1 / 2)
def test_dual_variables():
n = 5000 # nb bins
m = 6000 # nb bins
mean1 = 1000
mean2 = 1100
# bin positions
x = np.arange(n, dtype=np.float64)
y = np.arange(m, dtype=np.float64)
# Gaussian distributions
a = gauss(n, m=mean1, s=5) # m= mean, s= std
b = gauss(m, m=mean2, s=10)
# loss matrix
M = ot.dist(x.reshape((-1, 1)), y.reshape((-1, 1))) ** (1. / 2)
print('Computing {} EMD '.format(1))
# emd loss 1 proc
ot.tic()
G, log = ot.emd(a, b, M, log=True)
ot.toc('1 proc : {} s')
ot.tic()
G2 = ot.emd(b, a, np.ascontiguousarray(M.T))
ot.toc('1 proc : {} s')
cost1 = (G * M).sum()
# Check symmetry
np.testing.assert_array_almost_equal(cost1, (M * G2.T).sum())
# Check with closed-form solution for gaussians
np.testing.assert_almost_equal(cost1, np.abs(mean1 - mean2))
# Check that both cost computations are equivalent
np.testing.assert_almost_equal(cost1, log['cost'])
check_duality_gap(a, b, M, G, log['u'], log['v'], log['cost'])
def solve(fake_feature,true_feature):
# get the optimal matching between fake and true. assume #fake < # true
M=distance(fake_feature,true_feature,True)
emd = ot.emd([], [], M.numpy())
map= np.zeros(fake_feature.size(0))
for i in range(0,fake_feature.size(0)):
for j in range(0,true_feature.size(0)):
if emd[i][j]>0:
map[i]=j
return map
def jdot_krr(X,y,Xtest,gamma_g=1, numIterBCD = 10, alpha=1,lambd=1e1,
method='emd',reg=1,ktype='linear'):
# Initializations
n = X.shape[0]
ntest = Xtest.shape[0]
wa=np.ones((n,))/n
wb=np.ones((ntest,))/ntest
# original loss
C0=cdist(X,Xtest,metric='sqeuclidean')
#print np.max(C0)
C0=C0/np.median(C0)
# classifier
g = classif.KRRClassifier(lambd)
# compute kernels
if ktype=='rbf':
Kt=sklearn.metrics.pairwise.rbf_kernel(Xtest,Xtest,gamma=gamma_g)
else:
Kt=sklearn.metrics.pairwise.linear_kernel(Xtest,Xtest)
C = alpha*C0#+ cdist(y,ypred,metric='sqeuclidean')
k=0
while (k<numIterBCD):# and not changeLabels:
k=k+1
if method=='sinkhorn':
G = ot.sinkhorn(wa,wb,C,reg)
if method=='emd':
G= ot.emd(wa,wb,C)
Yst=ntest*G.T.dot(y)
g.fit(Kt,Yst)
ypred=g.predict(Kt)
# function cost
fcost = cdist(y,ypred,metric='sqeuclidean')
C=alpha*C0+fcost
return g,np.sum(G*(fcost))
def test_emd_empty():
# test emd and emd2 for simple identity
n = 100
rng = np.random.RandomState(0)
x = rng.randn(n, 2)
u = ot.utils.unif(n)
M = ot.dist(x, x)
G = ot.emd([], [], M)
# check G is identity
np.testing.assert_allclose(G, np.eye(n) / n)
# check constratints
np.testing.assert_allclose(u, G.sum(1)) # cf convergence sinkhorn
np.testing.assert_allclose(u, G.sum(0)) # cf convergence sinkhorn
w = ot.emd2([], [], M)
# check loss=0
np.testing.assert_allclose(w, 0)
def test_warnings():
n = 100 # nb bins
m = 100 # nb bins
mean1 = 30
mean2 = 50
# bin positions
x = np.arange(n, dtype=np.float64)
y = np.arange(m, dtype=np.float64)
# Gaussian distributions
a = gauss(n, m=mean1, s=5) # m= mean, s= std
b = gauss(m, m=mean2, s=10)
# loss matrix
M = ot.dist(x.reshape((-1, 1)), y.reshape((-1, 1))) ** (1. / 2)
print('Computing {} EMD '.format(1))
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
print('Computing {} EMD '.format(1))
ot.emd(a, b, M, numItermax=1)
assert "numItermax" in str(w[-1].message)
assert len(w) == 1
a[0] = 100
print('Computing {} EMD '.format(2))
ot.emd(a, b, M)
assert "infeasible" in str(w[-1].message)
assert len(w) == 2
a[0] = -1
print('Computing {} EMD '.format(2))
ot.emd(a, b, M)
assert "infeasible" in str(w[-1].message)
assert len(w) == 3
pl.xlabel('x')
pl.ylabel('y')
pl.legend()
pl.title('Toy regression example')
#%% TLOT
itermax=5
alpha=1
C0=cdist(xs,xt,metric='sqeuclidean')
#print np.max(C0)
C0=C0/np.median(C0)
fcost = cdist(ys,yt,metric='sqeuclidean')
C=alpha*C0+fcost
G0=ot.emd(ot.unif(n),ot.unif(n),C)
fit_params={'epochs':100}
model,loss = jdot.jdot_nn_l2(get_model,xs,ys,xt,ytest=yt,fit_params=fit_params,numIterBCD = itermax, alpha=alpha)
ypred=model.predict(xvisu.reshape((-1,1)))
pl.figure(2)
pl.clf()
pl.scatter(xs,ys,label='Source samples',edgecolors='k')
pl.scatter(xt,yt,label='Target samples',edgecolors='k')
pl.plot(xvisu,fs_s(xvisu),'b',label='Source model')
pl.plot(xvisu,fs_t(xvisu),'g',label='Target model')
pl.plot(xvisu,ypred,'r',label='JDOT model')
def jdot_nn_l2(get_model,X,Y,Xtest,ytest=[],fit_params={},reset_model=True, numIterBCD = 10, alpha=1,method='emd',reg=1,nb_epoch=100,batch_size=10):
# get model should return a new model compiled with l2 loss
# Initializations
n = X.shape[0]
ntest = Xtest.shape[0]
wa=np.ones((n,))/n
wb=np.ones((ntest,))/ntest
# original loss
C0=cdist(X,Xtest,metric='sqeuclidean')
C0=C0/np.max(C0)
# classifier
g = get_model()
TBR = []
sav_fcost = []
sav_totalcost = []
results = {}
#Init initial g(.)
g.fit(X,Y,**fit_params)
ypred=g.predict(Xtest)
C = alpha*C0+ cdist(Y,ypred,metric='sqeuclidean')
# do it only if the final labels were given
if len(ytest):
ydec=np.argmax(ypred,1)+1
TBR1=np.mean(ytest==ydec)
TBR.append(TBR1)
k=0
changeLabels=False
while (k<numIterBCD):# and not changeLabels:
k=k+1
if method=='sinkhorn':
G = ot.sinkhorn(wa,wb,C,reg)
if method=='emd':
G= ot.emd(wa,wb,C)
Yst=ntest*G.T.dot(Y)
if reset_model:
g=get_model()
g.fit(Xtest,Yst,**fit_params)
ypred=g.predict(Xtest)
# function cost
fcost = cdist(Y,ypred,metric='sqeuclidean')
#pl.figure()
#pl.imshow(fcost)
#pl.show()
C=alpha*C0+fcost
ydec_tmp=np.argmax(ypred,1)+1
if k>1:
changeLabels=np.all(ydec_tmp==ydec)
sav_fcost.append(np.sum(G*fcost))
sav_totalcost.append(np.sum(G*(alpha*C0+fcost)))
ydec=ydec_tmp
if len(ytest):
TBR1=np.mean((ytest-ypred)**2)
TBR.append(TBR1)
results['ypred0']=ypred
results['ypred']=np.argmax(ypred,1)+1
if len(ytest):
results['mse']=TBR
results['clf']=g
results['fcost']=sav_fcost
results['totalcost']=sav_totalcost
return g,results
pl.subplot(1, 3, 2)
pl.imshow(M2, interpolation='nearest')
pl.title('Squared Euclidean cost')
pl.subplot(1, 3, 3)
pl.imshow(Mp, interpolation='nearest')
pl.title('Sqrt Euclidean cost')
pl.tight_layout()
##############################################################################
# Dataset 1 : Plot OT Matrices
# ----------------------------
#%% EMD
G1 = ot.emd(a, b, M1)
G2 = ot.emd(a, b, M2)
Gp = ot.emd(a, b, Mp)
# OT matrices
pl.figure(3, figsize=(7, 3))
pl.subplot(1, 3, 1)
ot.plot.plot2D_samples_mat(xs, xt, G1, c=[.5, .5, 1])
pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
pl.axis('equal')
# pl.legend(loc=0)
pl.title('OT Euclidean')
pl.subplot(1, 3, 2)
def jdot_svm(X,y,Xtest,
ytest=[],gamma_g=1, numIterBCD = 10, alpha=1,
lambd=1e1, method='emd',reg_sink=1,ktype='linear'):
# Initializations
n = X.shape[0]
ntest = Xtest.shape[0]
wa=np.ones((n,))/n
wb=np.ones((ntest,))/ntest
# original loss
C0=cdist(X,Xtest,metric='sqeuclidean')
# classifier
g = classif.SVMClassifier(lambd)
# compute kernels
if ktype=='rbf':
Kt=sklearn.metrics.pairwise.rbf_kernel(Xtest,gamma=gamma_g)
#Ks=sklearn.metrics.pairwise.rbf_kernel(X,gamma=gamma_g)
else:
Kt=sklearn.metrics.pairwise.linear_kernel(Xtest)
#Ks=sklearn.metrics.pairwise.linear_kernel(X)
TBR = []
sav_fcost = []
sav_totalcost = []
results = {}
ypred=np.zeros(y.shape)
Chinge=np.zeros(C0.shape)
C=alpha*C0+Chinge
# do it only if the final labels were given
if len(ytest):
TBR.append(np.mean(ytest==np.argmax(ypred,1)+1))
k=0
while (k<numIterBCD):
k=k+1
if method=='sinkhorn':
G = ot.sinkhorn(wa,wb,C,reg_sink)
if method=='emd':
G= ot.emd(wa,wb,C)
if k>1:
sav_fcost.append(np.sum(G*Chinge))
sav_totalcost.append(np.sum(G*(alpha*C0+Chinge)))
Yst=ntest*G.T.dot((y+1)/2.)
#Yst=ntest*G.T.dot(y_f)
g.fit(Kt,Yst)
ypred=g.predict(Kt)
Chinge=classif.loss_hinge(y,ypred)
#Chinge=SVMclassifier.loss_hinge(y_f*2-1,ypred*2-1)
C=alpha*C0+Chinge
if len(ytest):
TBR1=np.mean(ytest==np.argmax(ypred,1)+1)
TBR.append(TBR1)
results['ypred']=np.argmax(ypred,1)+1
if len(ytest):
results['TBR']=TBR
results['clf']=g
results['G']=G
results['fcost']=sav_fcost
results['totalcost']=sav_totalcost
return g,results
pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
pl.legend(loc=0)
pl.title('Source and target distributions')
pl.figure(2)
pl.imshow(M, interpolation='nearest')
pl.title('Cost matrix M')
##############################################################################
# Compute EMD
# -----------
#%% EMD
G0 = ot.emd(a, b, M)
pl.figure(3)
pl.imshow(G0, interpolation='nearest')
pl.title('OT matrix G0')
pl.figure(4)
ot.plot.plot2D_samples_mat(xs, xt, G0, c=[.5, .5, 1])
pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
pl.legend(loc=0)
pl.title('OT matrix with samples')
##############################################################################
# Compute Sinkhorn