diff options
Diffstat (limited to 'ot')
| -rw-r--r-- | ot/__init__.py | 12 | ||||
| -rw-r--r-- | ot/da.py | 8 | ||||
| -rw-r--r-- | ot/gpu/__init__.py | 6 | ||||
| -rw-r--r-- | ot/gromov.py | 164 | ||||
| -rw-r--r-- | ot/lp/__init__.py | 47 | ||||
| -rw-r--r-- | ot/lp/emd_wrap.pyx | 11 |
6 files changed, 151 insertions, 97 deletions
diff --git a/ot/__init__.py b/ot/__init__.py index 3892d1d..89bdc78 100644 --- a/ot/__init__.py +++ b/ot/__init__.py @@ -1,6 +1,16 @@ -"""Python Optimal Transport toolbox +""" + +This is the main module of the POT toolbox. It provides easy access to +a number of functions described below. +.. warning:: + The list of automatically imported sub-modules is as follows: + :py:mod:`ot.lp`, :py:mod:`ot.bregman`, :py:mod:`ot.optim` + :py:mod:`ot.utils`, :py:mod:`ot.datasets`, + :py:mod:`ot.gromov`, :py:mod:`ot.smooth` + :py:mod:`ot.stochastic` + The other sub-modules are not imported due to additional dependencies. """ @@ -41,15 +41,15 @@ def sinkhorn_lpl1_mm(a, labels_a, b, M, reg, eta=0.1, numItermax=10, where : - M is the (ns,nt) metric cost matrix - - :math:`\Omega_e` is the entropic regularization term - :math:`\Omega_e(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` - - :math:`\Omega_g` is the group lasso regulaization term + - :math:`\Omega_e` is the entropic regularization term :math:`\Omega_e + (\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` + - :math:`\Omega_g` is the group lasso regularization term :math:`\Omega_g(\gamma)=\sum_{i,c} \|\gamma_{i,\mathcal{I}_c}\|^{1/2}_1` where :math:`\mathcal{I}_c` are the index of samples from class c in the source domain. - a and b are source and target weights (sum to 1) - The algorithm used for solving the problem is the generalised conditional + The algorithm used for solving the problem is the generalized conditional gradient as proposed in [5]_ [7]_ diff --git a/ot/gpu/__init__.py b/ot/gpu/__init__.py index deda6b1..6a2afcf 100644 --- a/ot/gpu/__init__.py +++ b/ot/gpu/__init__.py @@ -5,11 +5,15 @@ This module provides GPU implementation for several OT solvers and utility functions. The GPU backend in handled by `cupy <https://cupy.chainer.org/>`_. +.. warning:: + Note that by default the module is not import in :mod:`ot`. in order to + use it you need to import :mod:`ot.gpu` . + By default, the functions in this module accept and return numpy arrays in order to proide drop-in replacement for the other POT function but the transfer between CPU en GPU comes with a significant overhead. -In order to get the best erformances, we recommend to give only cupy +In order to get the best performances, we recommend to give only cupy arrays to the functions and desactivate the conversion to numpy of the result of the function with parameter ``to_numpy=False``. diff --git a/ot/gromov.py b/ot/gromov.py index ca96b31..cd961b0 100644 --- a/ot/gromov.py +++ b/ot/gromov.py @@ -72,8 +72,8 @@ def init_matrix(C1, C2, p, q, loss_fun='square_loss'): References
----------
.. [12] Peyré, Gabriel, Marco Cuturi, and Justin Solomon,
- "Gromov-Wasserstein averaging of kernel and distance matrices."
- International Conference on Machine Learning (ICML). 2016.
+ "Gromov-Wasserstein averaging of kernel and distance matrices."
+ International Conference on Machine Learning (ICML). 2016.
"""
@@ -137,8 +137,8 @@ def tensor_product(constC, hC1, hC2, T): References
----------
.. [12] Peyré, Gabriel, Marco Cuturi, and Justin Solomon,
- "Gromov-Wasserstein averaging of kernel and distance matrices."
- International Conference on Machine Learning (ICML). 2016.
+ "Gromov-Wasserstein averaging of kernel and distance matrices."
+ International Conference on Machine Learning (ICML). 2016.
"""
A = -np.dot(hC1, T).dot(hC2.T)
@@ -172,8 +172,8 @@ def gwloss(constC, hC1, hC2, T): References
----------
.. [12] Peyré, Gabriel, Marco Cuturi, and Justin Solomon,
- "Gromov-Wasserstein averaging of kernel and distance matrices."
- International Conference on Machine Learning (ICML). 2016.
+ "Gromov-Wasserstein averaging of kernel and distance matrices."
+ International Conference on Machine Learning (ICML). 2016.
"""
@@ -207,8 +207,8 @@ def gwggrad(constC, hC1, hC2, T): References
----------
.. [12] Peyré, Gabriel, Marco Cuturi, and Justin Solomon,
- "Gromov-Wasserstein averaging of kernel and distance matrices."
- International Conference on Machine Learning (ICML). 2016.
+ "Gromov-Wasserstein averaging of kernel and distance matrices."
+ International Conference on Machine Learning (ICML). 2016.
"""
return 2 * tensor_product(constC, hC1, hC2,
@@ -277,15 +277,15 @@ def gromov_wasserstein(C1, C2, p, q, loss_fun, log=False, armijo=False, **kwargs The function solves the following optimization problem:
.. math::
- \GW_Dist = \min_T \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}
+ GW = \min_T \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}
Where :
- C1 : Metric cost matrix in the source space
- C2 : Metric cost matrix in the target space
- p : distribution in the source space
- q : distribution in the target space
- L : loss function to account for the misfit between the similarity matrices
- H : entropy
+ - C1 : Metric cost matrix in the source space
+ - C2 : Metric cost matrix in the target space
+ - p : distribution in the source space
+ - q : distribution in the target space
+ - L : loss function to account for the misfit between the similarity matrices
+ - H : entropy
Parameters
----------
@@ -312,7 +312,7 @@ def gromov_wasserstein(C1, C2, p, q, loss_fun, log=False, armijo=False, **kwargs If True the steps of the line-search is found via an armijo research. Else closed form is used.
If there is convergence issues use False.
**kwargs : dict
- parameters can be directly pased to the ot.optim.cg solver
+ parameters can be directly passed to the ot.optim.cg solver
Returns
-------
@@ -355,25 +355,31 @@ def gromov_wasserstein(C1, C2, p, q, loss_fun, log=False, armijo=False, **kwargs def fused_gromov_wasserstein(M, C1, C2, p, q, loss_fun='square_loss', alpha=0.5, armijo=False, log=False, **kwargs):
"""
Computes the FGW transport between two graphs see [24]
+
.. math::
- \gamma = arg\min_\gamma (1-\alpha)*<\gamma,M>_F + alpha* \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}
+ \gamma = arg\min_\gamma (1-\alpha)*<\gamma,M>_F + \alpha* \sum_{i,j,k,l}
+ L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}
+
s.t. \gamma 1 = p
\gamma^T 1= q
\gamma\geq 0
+
where :
- M is the (ns,nt) metric cost matrix
- :math:`f` is the regularization term ( and df is its gradient)
- a and b are source and target weights (sum to 1)
- L is a loss function to account for the misfit between the similarity matrices
- The algorithm used for solving the problem is conditional gradient as discussed in [1]_
+
+ The algorithm used for solving the problem is conditional gradient as discussed in [24]_
+
Parameters
----------
M : ndarray, shape (ns, nt)
Metric cost matrix between features across domains
C1 : ndarray, shape (ns, ns)
- Metric cost matrix respresentative of the structure in the source space
+ Metric cost matrix representative of the structure in the source space
C2 : ndarray, shape (nt, nt)
- Metric cost matrix espresentative of the structure in the target space
+ Metric cost matrix representative of the structure in the target space
p : ndarray, shape (ns,)
distribution in the source space
q : ndarray, shape (nt,)
@@ -392,19 +398,23 @@ def fused_gromov_wasserstein(M, C1, C2, p, q, loss_fun='square_loss', alpha=0.5, If True the steps of the line-search is found via an armijo research. Else closed form is used.
If there is convergence issues use False.
**kwargs : dict
- parameters can be directly pased to the ot.optim.cg solver
+ parameters can be directly passed to the ot.optim.cg solver
+
Returns
-------
gamma : (ns x nt) ndarray
Optimal transportation matrix for the given parameters
log : dict
log dictionary return only if log==True in parameters
+
+
References
----------
.. [24] Vayer Titouan, Chapel Laetitia, Flamary R{\'e}mi, Tavenard Romain
- and Courty Nicolas
- "Optimal Transport for structured data with application on graphs"
- International Conference on Machine Learning (ICML). 2019.
+ and Courty Nicolas "Optimal Transport for structured data with
+ application on graphs", International Conference on Machine Learning
+ (ICML). 2019.
+
"""
constC, hC1, hC2 = init_matrix(C1, C2, p, q, loss_fun)
@@ -428,17 +438,23 @@ def fused_gromov_wasserstein(M, C1, C2, p, q, loss_fun='square_loss', alpha=0.5, def fused_gromov_wasserstein2(M, C1, C2, p, q, loss_fun='square_loss', alpha=0.5, armijo=False, log=False, **kwargs):
"""
Computes the FGW distance between two graphs see [24]
+
.. math::
- \gamma = arg\min_\gamma (1-\alpha)*<\gamma,M>_F + alpha* \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}
+ \min_\gamma (1-\alpha)*<\gamma,M>_F + \alpha* \sum_{i,j,k,l}
+ L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}
+
+
s.t. \gamma 1 = p
\gamma^T 1= q
\gamma\geq 0
+
where :
- M is the (ns,nt) metric cost matrix
- :math:`f` is the regularization term ( and df is its gradient)
- a and b are source and target weights (sum to 1)
- L is a loss function to account for the misfit between the similarity matrices
The algorithm used for solving the problem is conditional gradient as discussed in [1]_
+
Parameters
----------
M : ndarray, shape (ns, nt)
@@ -466,16 +482,18 @@ def fused_gromov_wasserstein2(M, C1, C2, p, q, loss_fun='square_loss', alpha=0.5 If there is convergence issues use False.
**kwargs : dict
parameters can be directly pased to the ot.optim.cg solver
+
Returns
-------
gamma : (ns x nt) ndarray
Optimal transportation matrix for the given parameters
log : dict
log dictionary return only if log==True in parameters
+
References
----------
.. [24] Vayer Titouan, Chapel Laetitia, Flamary R{\'e}mi, Tavenard Romain
- and Courty Nicolas
+ and Courty Nicolas
"Optimal Transport for structured data with application on graphs"
International Conference on Machine Learning (ICML). 2019.
"""
@@ -506,22 +524,22 @@ def gromov_wasserstein2(C1, C2, p, q, loss_fun, log=False, armijo=False, **kwarg The function solves the following optimization problem:
.. math::
- \GW_Dist = \min_T \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}
+ GW = \min_T \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}
Where :
- C1 : Metric cost matrix in the source space
- C2 : Metric cost matrix in the target space
- p : distribution in the source space
- q : distribution in the target space
- L : loss function to account for the misfit between the similarity matrices
- H : entropy
+ - C1 : Metric cost matrix in the source space
+ - C2 : Metric cost matrix in the target space
+ - p : distribution in the source space
+ - q : distribution in the target space
+ - L : loss function to account for the misfit between the similarity matrices
+ - H : entropy
Parameters
----------
C1 : ndarray, shape (ns, ns)
Metric cost matrix in the source space
C2 : ndarray, shape (nt, nt)
- Metric costfr matrix in the target space
+ Metric cost matrix in the target space
p : ndarray, shape (ns,)
distribution in the source space
q : ndarray, shape (nt,)
@@ -587,21 +605,21 @@ def entropic_gromov_wasserstein(C1, C2, p, q, loss_fun, epsilon, The function solves the following optimization problem:
.. math::
- \GW = arg\min_T \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}-\epsilon(H(T))
+ GW = arg\min_T \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}-\epsilon(H(T))
- s.t. \GW 1 = p
+ s.t. T 1 = p
- \GW^T 1= q
+ T^T 1= q
- \GW\geq 0
+ T\geq 0
Where :
- C1 : Metric cost matrix in the source space
- C2 : Metric cost matrix in the target space
- p : distribution in the source space
- q : distribution in the target space
- L : loss function to account for the misfit between the similarity matrices
- H : entropy
+ - C1 : Metric cost matrix in the source space
+ - C2 : Metric cost matrix in the target space
+ - p : distribution in the source space
+ - q : distribution in the target space
+ - L : loss function to account for the misfit between the similarity matrices
+ - H : entropy
Parameters
----------
@@ -629,14 +647,13 @@ def entropic_gromov_wasserstein(C1, C2, p, q, loss_fun, epsilon, Returns
-------
T : ndarray, shape (ns, nt)
- coupling between the two spaces that minimizes :
- \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}-\epsilon(H(T))
+ Optimal coupling between the two spaces
References
----------
.. [12] Peyré, Gabriel, Marco Cuturi, and Justin Solomon,
- "Gromov-Wasserstein averaging of kernel and distance matrices."
- International Conference on Machine Learning (ICML). 2016.
+ "Gromov-Wasserstein averaging of kernel and distance matrices."
+ International Conference on Machine Learning (ICML). 2016.
"""
@@ -695,15 +712,15 @@ def entropic_gromov_wasserstein2(C1, C2, p, q, loss_fun, epsilon, The function solves the following optimization problem:
.. math::
- \GW_Dist = \min_T \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}-\epsilon(H(T))
+ GW = \min_T \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}-\epsilon(H(T))
Where :
- C1 : Metric cost matrix in the source space
- C2 : Metric cost matrix in the target space
- p : distribution in the source space
- q : distribution in the target space
- L : loss function to account for the misfit between the similarity matrices
- H : entropy
+ - C1 : Metric cost matrix in the source space
+ - C2 : Metric cost matrix in the target space
+ - p : distribution in the source space
+ - q : distribution in the target space
+ - L : loss function to account for the misfit between the similarity matrices
+ - H : entropy
Parameters
----------
@@ -736,8 +753,8 @@ def entropic_gromov_wasserstein2(C1, C2, p, q, loss_fun, epsilon, References
----------
.. [12] Peyré, Gabriel, Marco Cuturi, and Justin Solomon,
- "Gromov-Wasserstein averaging of kernel and distance matrices."
- International Conference on Machine Learning (ICML). 2016.
+ "Gromov-Wasserstein averaging of kernel and distance matrices."
+ International Conference on Machine Learning (ICML). 2016.
"""
@@ -762,13 +779,13 @@ def entropic_gromov_barycenters(N, Cs, ps, p, lambdas, loss_fun, epsilon, The function solves the following optimization problem:
.. math::
- C = argmin_C\in R^{NxN} \sum_s \lambda_s GW(C,Cs,p,ps)
+ C = argmin_{C\in R^{NxN}} \sum_s \lambda_s GW(C,C_s,p,p_s)
Where :
- Cs : metric cost matrix
- ps : distribution
+ - :math:`C_s` : metric cost matrix
+ - :math:`p_s` : distribution
Parameters
----------
@@ -806,8 +823,8 @@ def entropic_gromov_barycenters(N, Cs, ps, p, lambdas, loss_fun, epsilon, References
----------
.. [12] Peyré, Gabriel, Marco Cuturi, and Justin Solomon,
- "Gromov-Wasserstein averaging of kernel and distance matrices."
- International Conference on Machine Learning (ICML). 2016.
+ "Gromov-Wasserstein averaging of kernel and distance matrices."
+ International Conference on Machine Learning (ICML). 2016.
"""
@@ -876,8 +893,8 @@ def gromov_barycenters(N, Cs, ps, p, lambdas, loss_fun, Where :
- Cs : metric cost matrix
- ps : distribution
+ - Cs : metric cost matrix
+ - ps : distribution
Parameters
----------
@@ -913,8 +930,8 @@ def gromov_barycenters(N, Cs, ps, p, lambdas, loss_fun, References
----------
.. [12] Peyré, Gabriel, Marco Cuturi, and Justin Solomon,
- "Gromov-Wasserstein averaging of kernel and distance matrices."
- International Conference on Machine Learning (ICML). 2016.
+ "Gromov-Wasserstein averaging of kernel and distance matrices."
+ International Conference on Machine Learning (ICML). 2016.
"""
@@ -972,6 +989,8 @@ def fgw_barycenters(N, Ys, Cs, ps, lambdas, alpha, fixed_structure=False, fixed_ verbose=False, log=False, init_C=None, init_X=None):
"""
Compute the fgw barycenter as presented eq (5) in [24].
+
+ Parameters
----------
N : integer
Desired number of samples of the target barycenter
@@ -993,8 +1012,9 @@ def fgw_barycenters(N, Ys, Cs, ps, lambdas, alpha, fixed_structure=False, fixed_ initialization for the barycenters' structure matrix. If not set random init
init_X : ndarray, shape (N,d), optional
initialization for the barycenters' features. If not set random init
+
Returns
- ----------
+ -------
X : ndarray, shape (N,d)
Barycenters' features
C : ndarray, shape (N,N)
@@ -1002,11 +1022,13 @@ def fgw_barycenters(N, Ys, Cs, ps, lambdas, alpha, fixed_structure=False, fixed_ log_: dictionary
Only returned when log=True
T : list of (N,ns) transport matrices
- Ms : all distance matrices between the feature of the barycenter and the other features dist(X,Ys) shape (N,ns)
+ Ms : all distance matrices between the feature of the barycenter and the
+ other features dist(X,Ys) shape (N,ns)
+
References
----------
.. [24] Vayer Titouan, Chapel Laetitia, Flamary R{\'e}mi, Tavenard Romain
- and Courty Nicolas
+ and Courty Nicolas
"Optimal Transport for structured data with application on graphs"
International Conference on Machine Learning (ICML). 2019.
"""
@@ -1107,6 +1129,7 @@ def update_sructure_matrix(p, lambdas, T, Cs): """
Updates C according to the L2 Loss kernel with the S Ts couplings
calculated at each iteration
+
Parameters
----------
p : ndarray, shape (N,)
@@ -1117,6 +1140,7 @@ def update_sructure_matrix(p, lambdas, T, Cs): the S Ts couplings calculated at each iteration
Cs : list of S ndarray, shape(ns,ns)
Metric cost matrices
+
Returns
----------
C : ndarray, shape (nt,nt)
@@ -1132,6 +1156,7 @@ def update_feature_matrix(lambdas, Ys, Ts, p): """
Updates the feature with respect to the S Ts couplings. See "Solving the barycenter problem with Block Coordinate Descent (BCD)" in [24]
calculated at each iteration
+
Parameters
----------
p : ndarray, shape (N,)
@@ -1142,6 +1167,7 @@ def update_feature_matrix(lambdas, Ys, Ts, p): the S Ts couplings calculated at each iteration
Ys : list of S ndarray, shape(d,ns)
The features
+
Returns
----------
X : ndarray, shape (d,N)
diff --git a/ot/lp/__init__.py b/ot/lp/__init__.py index 02cbd8c..ed6fa52 100644 --- a/ot/lp/__init__.py +++ b/ot/lp/__init__.py @@ -1,6 +1,9 @@ # -*- coding: utf-8 -*- """ Solvers for the original linear program OT problem + + + """ # Author: Remi Flamary <remi.flamary@unice.fr> @@ -37,26 +40,30 @@ def emd(a, b, M, numItermax=100000, log=False): - M is the metric cost matrix - a and b are the sample weights + .. warning:: + Note that the M matrix needs to be a C-order numpy.array in float64 + format. + Uses the algorithm proposed in [1]_ Parameters ---------- - a : (ns,) ndarray, float64 - Source histogram (uniform weigth if empty list) - b : (nt,) ndarray, float64 - Target histogram (uniform weigth if empty list) - M : (ns,nt) ndarray, float64 - loss matrix + a : (ns,) numpy.ndarray, float64 + Source histogram (uniform weight if empty list) + b : (nt,) numpy.ndarray, float64 + Target histogram (uniform weight if empty list) + M : (ns,nt) numpy.ndarray, float64 + Loss matrix (c-order array with type float64) numItermax : int, optional (default=100000) The maximum number of iterations before stopping the optimization algorithm if it has not converged. - log: boolean, optional (default=False) + log: bool, optional (default=False) If True, returns a dictionary containing the cost and dual variables. Otherwise returns only the optimal transportation matrix. Returns ------- - gamma: (ns x nt) ndarray + gamma: (ns x nt) numpy.ndarray Optimal transportation matrix for the given parameters log: dict If input log is true, a dictionary containing the cost and dual @@ -128,16 +135,20 @@ def emd2(a, b, M, processes=multiprocessing.cpu_count(), - M is the metric cost matrix - a and b are the sample weights + .. warning:: + Note that the M matrix needs to be a C-order numpy.array in float64 + format. + Uses the algorithm proposed in [1]_ Parameters ---------- - a : (ns,) ndarray, float64 - Source histogram (uniform weigth if empty list) - b : (nt,) ndarray, float64 - Target histogram (uniform weigth if empty list) - M : (ns,nt) ndarray, float64 - loss matrix + a : (ns,) numpy.ndarray, float64 + Source histogram (uniform weight if empty list) + b : (nt,) numpy.ndarray, float64 + Target histogram (uniform weight if empty list) + M : (ns,nt) numpy.ndarray, float64 + Loss matrix (c-order array with type float64) numItermax : int, optional (default=100000) The maximum number of iterations before stopping the optimization algorithm if it has not converged. @@ -151,7 +162,7 @@ def emd2(a, b, M, processes=multiprocessing.cpu_count(), ------- gamma: (ns x nt) ndarray Optimal transportation matrix for the given parameters - log: dict + log: dictnp If input log is true, a dictionary containing the cost and dual variables and exit status @@ -231,9 +242,9 @@ def free_support_barycenter(measures_locations, measures_weights, X_init, b=None Parameters ---------- - measures_locations : list of (k_i,d) np.ndarray + measures_locations : list of (k_i,d) numpy.ndarray The discrete support of a measure supported on k_i locations of a d-dimensional space (k_i can be different for each element of the list) - measures_weights : list of (k_i,) np.ndarray + measures_weights : list of (k_i,) numpy.ndarray Numpy arrays where each numpy array has k_i non-negatives values summing to one representing the weights of each discrete input measure X_init : (k,d) np.ndarray @@ -246,7 +257,7 @@ def free_support_barycenter(measures_locations, measures_weights, X_init, b=None numItermax : int, optional Max number of iterations stopThr : float, optional - Stop threshol on error (>0) + Stop threshold on error (>0) verbose : bool, optional Print information along iterations log : bool, optional diff --git a/ot/lp/emd_wrap.pyx b/ot/lp/emd_wrap.pyx index 83ee6aa..edb5f7c 100644 --- a/ot/lp/emd_wrap.pyx +++ b/ot/lp/emd_wrap.pyx @@ -55,13 +55,16 @@ def emd_c(np.ndarray[double, ndim=1, mode="c"] a, np.ndarray[double, ndim=1, mod - M is the metric cost matrix - a and b are the sample weights + .. warning:: + Note that the M matrix needs to be a C-order :py.cls:`numpy.array` + Parameters ---------- - a : (ns,) ndarray, float64 + a : (ns,) numpy.ndarray, float64 source histogram - b : (nt,) ndarray, float64 + b : (nt,) numpy.ndarray, float64 target histogram - M : (ns,nt) ndarray, float64 + M : (ns,nt) numpy.ndarray, float64 loss matrix max_iter : int The maximum number of iterations before stopping the optimization @@ -70,7 +73,7 @@ def emd_c(np.ndarray[double, ndim=1, mode="c"] a, np.ndarray[double, ndim=1, mod Returns ------- - gamma: (ns x nt) ndarray + gamma: (ns x nt) numpy.ndarray Optimal transportation matrix for the given parameters """ |