ot.gromov¶
GromovWasserstein and FusedGromovWasserstein solvers
Functions¶

ot.gromov.
entropic_gromov_barycenters
(N, Cs, ps, p, lambdas, loss_fun, epsilon, max_iter=1000, tol=1e09, verbose=False, log=False, init_C=None)[source]¶ Returns the gromovwasserstein barycenters of S measured similarity matrices
(Cs)_{s=1}^{s=S}
The function solves the following optimization problem:
\[C = argmin_{C\in R^{NxN}} \sum_s \lambda_s GW(C,C_s,p,p_s)\]Where :
\(C_s\) : metric cost matrix
\(p_s\) : distribution
 Parameters
N (int) – Size of the targeted barycenter
Cs (list of S np.ndarray of shape (ns,ns)) – Metric cost matrices
ps (list of S np.ndarray of shape (ns,)) – Sample weights in the S spaces
p (ndarray, shape(N,)) – Weights in the targeted barycenter
lambdas (list of float) – List of the S spaces’ weights.
loss_fun (callable) – Tensormatrix multiplication function based on specific loss function.
update (callable) – function(p,lambdas,T,Cs) that updates C according to a specific Kernel with the S Ts couplings calculated at each iteration
epsilon (float) – Regularization term >0
max_iter (int, optional) – Max number of iterations
tol (float, optional) – Stop threshol on error (>0)
verbose (bool, optional) – Print information along iterations.
log (bool, optional) – Record log if True.
init_C (bool  ndarray, shape (N, N)) – Random initial value for the C matrix provided by user.
 Returns
C – Similarity matrix in the barycenter space (permutated arbitrarily)
 Return type
ndarray, shape (N, N)
References
 12
Peyré, Gabriel, Marco Cuturi, and Justin Solomon, “GromovWasserstein averaging of kernel and distance matrices.” International Conference on Machine Learning (ICML). 2016.

ot.gromov.
entropic_gromov_wasserstein
(C1, C2, p, q, loss_fun, epsilon, max_iter=1000, tol=1e09, verbose=False, log=False)[source]¶ Returns the gromovwasserstein transport between (C1,p) and (C2,q)
(C1,p) and (C2,q)
The function solves the following optimization problem:
\[ \begin{align}\begin{aligned}GW = arg\min_T \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}\epsilon(H(T))\\s.t. T 1 = p\\ T^T 1= q\\ T\geq 0\end{aligned}\end{align} \]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
 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
p (ndarray, shape (ns,)) – Distribution in the source space
q (ndarray, shape (nt,)) – Distribution in the target space
loss_fun (string) – Loss function used for the solver either ‘square_loss’ or ‘kl_loss’
epsilon (float) – Regularization term >0
max_iter (int, optional) – Max number of iterations
tol (float, optional) – Stop threshold on error (>0)
verbose (bool, optional) – Print information along iterations
log (bool, optional) – Record log if True.
 Returns
T – Optimal coupling between the two spaces
 Return type
ndarray, shape (ns, nt)
References
 12
Peyré, Gabriel, Marco Cuturi, and Justin Solomon, “GromovWasserstein averaging of kernel and distance matrices.” International Conference on Machine Learning (ICML). 2016.
Examples using ot.gromov.entropic_gromov_wasserstein
¶

ot.gromov.
entropic_gromov_wasserstein2
(C1, C2, p, q, loss_fun, epsilon, max_iter=1000, tol=1e09, verbose=False, log=False)[source]¶ Returns the entropic gromovwasserstein discrepancy between the two measured similarity matrices
(C1,p) and (C2,q)
The function solves the following optimization problem:
\[GW = \min_T \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}\epsilon(H(T))\]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
 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
p (ndarray, shape (ns,)) – Distribution in the source space
q (ndarray, shape (nt,)) – Distribution in the target space
loss_fun (str) – Loss function used for the solver either ‘square_loss’ or ‘kl_loss’
epsilon (float) – Regularization term >0
max_iter (int, optional) – Max number of iterations
tol (float, optional) – Stop threshold on error (>0)
verbose (bool, optional) – Print information along iterations
log (bool, optional) – Record log if True.
 Returns
gw_dist – GromovWasserstein distance
 Return type
References
 12
Peyré, Gabriel, Marco Cuturi, and Justin Solomon, “GromovWasserstein averaging of kernel and distance matrices.” International Conference on Machine Learning (ICML). 2016.

ot.gromov.
fgw_barycenters
(N, Ys, Cs, ps, lambdas, alpha, fixed_structure=False, fixed_features=False, p=None, loss_fun='square_loss', max_iter=100, tol=1e09, verbose=False, log=False, init_C=None, init_X=None)[source]¶ Compute the fgw barycenter as presented eq (5) in [24].
 Parameters
N (integer) – Desired number of samples of the target barycenter
Ys (list of ndarray, each element has shape (ns,d)) – Features of all samples
Cs (list of ndarray, each element has shape (ns,ns)) – Structure matrices of all samples
ps (list of ndarray, each element has shape (ns,)) – Masses of all samples.
lambdas (list of float) – List of the S spaces’ weights
alpha (float) – Alpha parameter for the fgw distance
fixed_structure (bool) – Whether to fix the structure of the barycenter during the updates
fixed_features (bool) – Whether to fix the feature of the barycenter during the updates
loss_fun (str) – Loss function used for the solver either ‘square_loss’ or ‘kl_loss’
max_iter (int, optional) – Max number of iterations
tol (float, optional) – Stop threshol on error (>0).
verbose (bool, optional) – Print information along iterations.
log (bool, optional) – Record log if True.
init_C (ndarray, shape (N,N), optional) – Initialization for the barycenters’ structure matrix. If not set a random init is used.
init_X (ndarray, shape (N,d), optional) – Initialization for the barycenters’ features. If not set a random init is used.
 Returns
X (ndarray, shape (N, d)) – Barycenters’ features
C (ndarray, shape (N, N)) – Barycenters’ structure matrix
log_ (dict) – Only returned when log=True. It contains the keys: 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)
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.
Examples using ot.gromov.fgw_barycenters
¶

ot.gromov.
fused_gromov_wasserstein
(M, C1, C2, p, q, loss_fun='square_loss', alpha=0.5, armijo=False, log=False, **kwargs)[source]¶ Computes the FGW transport between two graphs see [24]
\[ \begin{align}\begin{aligned}\gamma = arg\min_\gamma (1\alpha)*<\gamma,M>_F + \alpha* \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}\\s.t. \gamma 1 = p \gamma^T 1= q \gamma\geq 0\end{aligned}\end{align} \]where :  M is the (ns,nt) metric cost matrix  p and q 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 [24]_
 Parameters
M (ndarray, shape (ns, nt)) – Metric cost matrix between features across domains
C1 (ndarray, shape (ns, ns)) – Metric cost matrix representative of the structure in the source space
C2 (ndarray, shape (nt, nt)) – Metric cost matrix representative of the structure in the target space
p (ndarray, shape (ns,)) – Distribution in the source space
q (ndarray, shape (nt,)) – Distribution in the target space
loss_fun (str, optional) – Loss function used for the solver
alpha (float, optional) – Tradeoff parameter (0 < alpha < 1)
armijo (bool, optional) – If True the steps of the linesearch is found via an armijo research. Else closed form is used. If there is convergence issues use False.
log (bool, optional) – record log if True
**kwargs (dict) – parameters can be directly passed to the ot.optim.cg solver
 Returns
gamma (ndarray, shape (ns, nt)) – 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.
Examples using ot.gromov.fused_gromov_wasserstein
¶

ot.gromov.
fused_gromov_wasserstein2
(M, C1, C2, p, q, loss_fun='square_loss', alpha=0.5, armijo=False, log=False, **kwargs)[source]¶ Computes the FGW distance between two graphs see [24]
\[ \begin{align}\begin{aligned}\min_\gamma (1\alpha)*<\gamma,M>_F + \alpha* \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}\\s.t. \gamma 1 = p \gamma^T 1= q \gamma\geq 0\end{aligned}\end{align} \]where :  M is the (ns,nt) metric cost matrix  p and q 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)) – Metric cost matrix between features across domains
C1 (ndarray, shape (ns, ns)) – Metric cost matrix respresentative of the structure in the source space.
C2 (ndarray, shape (nt, nt)) – Metric cost matrix espresentative of the structure in the target space.
p (ndarray, shape (ns,)) – Distribution in the source space.
q (ndarray, shape (nt,)) – Distribution in the target space.
loss_fun (str, optional) – Loss function used for the solver.
alpha (float, optional) – Tradeoff parameter (0 < alpha < 1)
armijo (bool, optional) – If True the steps of the linesearch is found via an armijo research. Else closed form is used. If there is convergence issues use False.
log (bool, optional) – Record log if True.
**kwargs (dict) – Parameters can be directly pased to the ot.optim.cg solver.
 Returns
gamma (ndarray, shape (ns, nt)) – 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.

ot.gromov.
gromov_barycenters
(N, Cs, ps, p, lambdas, loss_fun, max_iter=1000, tol=1e09, verbose=False, log=False, init_C=None)[source]¶ Returns the gromovwasserstein barycenters of S measured similarity matrices
(Cs)_{s=1}^{s=S}
The function solves the following optimization problem with block coordinate descent:
\[C = argmin_C\in R^NxN \sum_s \lambda_s GW(C,Cs,p,ps)\]Where :
Cs : metric cost matrix
ps : distribution
 Parameters
N (int) – Size of the targeted barycenter
Cs (list of S np.ndarray of shape (ns, ns)) – Metric cost matrices
ps (list of S np.ndarray of shape (ns,)) – Sample weights in the S spaces
p (ndarray, shape (N,)) – Weights in the targeted barycenter
lambdas (list of float) – List of the S spaces’ weights
loss_fun (tensormatrix multiplication function based on specific loss function) –
update (function(p,lambdas,T,Cs) that updates C according to a specific Kernel) – with the S Ts couplings calculated at each iteration
max_iter (int, optional) – Max number of iterations
tol (float, optional) – Stop threshol on error (>0).
verbose (bool, optional) – Print information along iterations.
log (bool, optional) – Record log if True.
init_C (bool  ndarray, shape(N,N)) – Random initial value for the C matrix provided by user.
 Returns
C – Similarity matrix in the barycenter space (permutated arbitrarily)
 Return type
ndarray, shape (N, N)
References
 12
Peyré, Gabriel, Marco Cuturi, and Justin Solomon, “GromovWasserstein averaging of kernel and distance matrices.” International Conference on Machine Learning (ICML). 2016.
Examples using ot.gromov.gromov_barycenters
¶

ot.gromov.
gromov_wasserstein
(C1, C2, p, q, loss_fun, log=False, armijo=False, **kwargs)[source]¶ Returns the gromovwasserstein transport between (C1,p) and (C2,q)
The function solves the following optimization problem:
\[GW = \min_T \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}\]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
 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
p (ndarray, shape (ns,)) – Distribution in the source space
q (ndarray, shape (nt,)) – Distribution in the target space
loss_fun (str) – loss function used for the solver either ‘square_loss’ or ‘kl_loss’
max_iter (int, optional) – Max number of iterations
tol (float, optional) – Stop threshold on error (>0)
verbose (bool, optional) – Print information along iterations
log (bool, optional) – record log if True
armijo (bool, optional) – If True the steps of the linesearch is found via an armijo research. Else closed form is used. If there is convergence issues use False.
**kwargs (dict) – parameters can be directly passed to the ot.optim.cg solver
 Returns
T (ndarray, shape (ns, nt)) –
 Doupling between the two spaces that minimizes:
sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}
log (dict) – Convergence information and loss.
References
 12
Peyré, Gabriel, Marco Cuturi, and Justin Solomon, “GromovWasserstein averaging of kernel and distance matrices.” International Conference on Machine Learning (ICML). 2016.
 13
Mémoli, Facundo. Gromov–Wasserstein distances and the metric approach to object matching. Foundations of computational mathematics 11.4 (2011): 417487.
Examples using ot.gromov.gromov_wasserstein
¶

ot.gromov.
gromov_wasserstein2
(C1, C2, p, q, loss_fun, log=False, armijo=False, **kwargs)[source]¶ Returns the gromovwasserstein discrepancy between (C1,p) and (C2,q)
The function solves the following optimization problem:
\[GW = \min_T \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}\]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
 Parameters
C1 (ndarray, shape (ns, ns)) – Metric cost matrix in the source space
C2 (ndarray, shape (nt, nt)) – Metric cost matrix in the target space
p (ndarray, shape (ns,)) – Distribution in the source space.
q (ndarray, shape (nt,)) – Distribution in the target space.
loss_fun (str) – loss function used for the solver either ‘square_loss’ or ‘kl_loss’
max_iter (int, optional) – Max number of iterations
tol (float, optional) – Stop threshold on error (>0)
verbose (bool, optional) – Print information along iterations
log (bool, optional) – record log if True
armijo (bool, optional) – If True the steps of the linesearch is found via an armijo research. Else closed form is used. If there is convergence issues use False.
 Returns
gw_dist (float) – GromovWasserstein distance
log (dict) – convergence information and Coupling marix
References
 12
Peyré, Gabriel, Marco Cuturi, and Justin Solomon, “GromovWasserstein averaging of kernel and distance matrices.” International Conference on Machine Learning (ICML). 2016.
 13
Mémoli, Facundo. Gromov–Wasserstein distances and the metric approach to object matching. Foundations of computational mathematics 11.4 (2011): 417487.

ot.gromov.
gwggrad
(constC, hC1, hC2, T)[source]¶ Return the gradient for GromovWasserstein
The gradient is computed as described in Proposition 2 in [12].
 Parameters
constC (ndarray, shape (ns, nt)) – Constant C matrix in Eq. (6)
hC1 (ndarray, shape (ns, ns)) – h1(C1) matrix in Eq. (6)
hC2 (ndarray, shape (nt, nt)) – h2(C) matrix in Eq. (6)
T (ndarray, shape (ns, nt)) – Current value of transport matrix T
 Returns
grad – Gromov Wasserstein gradient
 Return type
ndarray, shape (ns, nt)
References
 12
Peyré, Gabriel, Marco Cuturi, and Justin Solomon, “GromovWasserstein averaging of kernel and distance matrices.” International Conference on Machine Learning (ICML). 2016.

ot.gromov.
gwloss
(constC, hC1, hC2, T)[source]¶ Return the Loss for GromovWasserstein
The loss is computed as described in Proposition 1 Eq. (6) in [12].
 Parameters
constC (ndarray, shape (ns, nt)) – Constant C matrix in Eq. (6)
hC1 (ndarray, shape (ns, ns)) – h1(C1) matrix in Eq. (6)
hC2 (ndarray, shape (nt, nt)) – h2(C) matrix in Eq. (6)
T (ndarray, shape (ns, nt)) – Current value of transport matrix T
 Returns
loss – Gromov Wasserstein loss
 Return type
References
 12
Peyré, Gabriel, Marco Cuturi, and Justin Solomon, “GromovWasserstein averaging of kernel and distance matrices.” International Conference on Machine Learning (ICML). 2016.

ot.gromov.
init_matrix
(C1, C2, p, q, loss_fun='square_loss')[source]¶ Return loss matrices and tensors for GromovWasserstein fast computation
Returns the value of mathcal{L}(C1,C2) otimes T with the selected loss function as the loss function of GromowWasserstein discrepancy.
The matrices are computed as described in Proposition 1 in [12]
 Where :
C1 : Metric cost matrix in the source space
C2 : Metric cost matrix in the target space
T : A coupling between those two spaces
 The squareloss function L(a,b)=ab^2 is read as :
 L(a,b) = f1(a)+f2(b)h1(a)*h2(b) with :
f1(a)=(a^2)
f2(b)=(b^2)
h1(a)=a
h2(b)=2*b
 The klloss function L(a,b)=a*log(a/b)a+b is read as :
 L(a,b) = f1(a)+f2(b)h1(a)*h2(b) with :
f1(a)=a*log(a)a
f2(b)=b
h1(a)=a
h2(b)=log(b)
 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
T (ndarray, shape (ns, nt)) – Coupling between source and target spaces
p (ndarray, shape (ns,)) –
 Returns
constC (ndarray, shape (ns, nt)) – Constant C matrix in Eq. (6)
hC1 (ndarray, shape (ns, ns)) – h1(C1) matrix in Eq. (6)
hC2 (ndarray, shape (nt, nt)) – h2(C) matrix in Eq. (6)
References
 12
Peyré, Gabriel, Marco Cuturi, and Justin Solomon, “GromovWasserstein averaging of kernel and distance matrices.” International Conference on Machine Learning (ICML). 2016.

ot.gromov.
tensor_product
(constC, hC1, hC2, T)[source]¶ Return the tensor for GromovWasserstein fast computation
The tensor is computed as described in Proposition 1 Eq. (6) in [12].
 Parameters
constC (ndarray, shape (ns, nt)) – Constant C matrix in Eq. (6)
hC1 (ndarray, shape (ns, ns)) – h1(C1) matrix in Eq. (6)
hC2 (ndarray, shape (nt, nt)) – h2(C) matrix in Eq. (6)
 Returns
tens – mathcal{L}(C1,C2) otimes T tensormatrix multiplication result
 Return type
ndarray, shape (ns, nt)
References
 12
Peyré, Gabriel, Marco Cuturi, and Justin Solomon, “GromovWasserstein averaging of kernel and distance matrices.” International Conference on Machine Learning (ICML). 2016.

ot.gromov.
update_feature_matrix
(lambdas, Ys, Ts, p)[source]¶ 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,)) – masses in the targeted barycenter
lambdas (list of float) – List of the S spaces’ weights
Ts (list of S np.ndarray(ns,N)) – the S Ts couplings calculated at each iteration
Ys (list of S ndarray, shape(d,ns)) – The features.
 Returns
X
 Return type
ndarray, shape (d, N)
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.

ot.gromov.
update_kl_loss
(p, lambdas, T, Cs)[source]¶ Updates C according to the KL Loss kernel with the S Ts couplings calculated at each iteration
 Parameters
p (ndarray, shape (N,)) – Weights in the targeted barycenter.
lambdas (list of the S spaces' weights) –
T (list of S np.ndarray of shape (ns,N)) – The S Ts couplings calculated at each iteration.
Cs (list of S ndarray, shape(ns,ns)) – Metric cost matrices.
 Returns
C – updated C matrix
 Return type
ndarray, shape (ns,ns)

ot.gromov.
update_square_loss
(p, lambdas, T, Cs)[source]¶ Updates C according to the L2 Loss kernel with the S Ts couplings calculated at each iteration
 Parameters
p (ndarray, shape (N,)) – Masses in the targeted barycenter.
lambdas (list of float) – List of the S spaces’ weights.
T (list of S np.ndarray of shape (ns,N)) – The S Ts couplings calculated at each iteration.
Cs (list of S ndarray, shape(ns,ns)) – Metric cost matrices.
 Returns
C – Updated C matrix.
 Return type
ndarray, shape (nt, nt)

ot.gromov.
update_sructure_matrix
(p, lambdas, T, Cs)[source]¶ Updates C according to the L2 Loss kernel with the S Ts couplings.
It is calculated at each iteration
 Parameters
p (ndarray, shape (N,)) – Masses in the targeted barycenter.
lambdas (list of float) – List of the S spaces’ weights.
T (list of S ndarray of shape (ns, N)) – The S Ts couplings calculated at each iteration.
Cs (list of S ndarray, shape (ns, ns)) – Metric cost matrices.
 Returns
C – Updated C matrix.
 Return type
ndarray, shape (nt, nt)