ot.da

Domain adaptation with optimal transport

Functions

ot.da.OT_mapping_linear(xs, xt, reg=1e-06, ws=None, wt=None, bias=True, log=False)[source]

Return OT linear operator between samples.

The function estimates the optimal linear operator that aligns the two empirical distributions. This is equivalent to estimating the closed form mapping between two Gaussian distributions \(\mathcal{N}(\mu_s,\Sigma_s)\) and \(\mathcal{N}(\mu_t,\Sigma_t)\) as proposed in [14] and discussed in remark 2.29 in [15].

The linear operator from source to target \(M\)

\[M(\mathbf{x})= \mathbf{A} \mathbf{x} + \mathbf{b}\]

where :

\[ \begin{align}\begin{aligned}\mathbf{A} &= \Sigma_s^{-1/2} \left(\Sigma_s^{1/2}\Sigma_t\Sigma_s^{1/2} \right)^{1/2} \Sigma_s^{-1/2}\\\mathbf{b} &= \mu_t - \mathbf{A} \mu_s\end{aligned}\end{align} \]

Parameters

xs (array-like (ns,d)) – samples in the source domain
xt (array-like (nt,d)) – samples in the target domain
reg (float,optional) – regularization added to the diagonals of covariances (>0)
ws (array-like (ns,1), optional) – weights for the source samples
wt (array-like (ns,1), optional) – weights for the target samples
bias (boolean, optional) – estimate bias \(\mathbf{b}\) else \(\mathbf{b} = 0\) (default:True)
log (bool, optional) – record log if True

Returns

A ((d, d) array-like) – Linear operator
b ((1, d) array-like) – bias
log (dict) – log dictionary return only if log==True in parameters

References

14: Knott, M. and Smith, C. S. “On the optimal mapping of distributions”, Journal of Optimization Theory and Applications Vol 43, 1984
15: Peyré, G., & Cuturi, M. (2017). “Computational Optimal Transport”, 2018.

Examples using `ot.da.OT_mapping_linear`

ot.da.distribution_estimation_uniform(X)[source]

estimates a uniform distribution from an array of samples \(\mathbf{X}\)

Parameters: X (array-like, shape (n_samples, n_features)) – The array of samples
Returns: mu – The uniform distribution estimated from \(\mathbf{X}\)
Return type: array-like, shape (n_samples,)

ot.da.emd_laplace(a, b, xs, xt, M, sim='knn', sim_param=None, reg='pos', eta=1, alpha=0.5, numItermax=100, stopThr=1e-09, numInnerItermax=100000, stopInnerThr=1e-09, log=False, verbose=False)[source]

Solve the optimal transport problem (OT) with Laplacian regularization

\[ \begin{align}\begin{aligned}\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F + \eta \cdot \Omega_\alpha(\gamma)\\s.t. \ \gamma \mathbf{1} = \mathbf{a}\\ \gamma^T \mathbf{1} = \mathbf{b}\\ \gamma \geq 0\end{aligned}\end{align} \]

where:

\(\mathbf{a}\) and \(\mathbf{b}\) are source and target weights (sum to 1)
\(\mathbf{x_s}\) and \(\mathbf{x_t}\) are source and target samples
\(\mathbf{M}\) is the (ns, nt) metric cost matrix
\(\Omega_\alpha\) is the Laplacian regularization term

\[\Omega_\alpha = \frac{1 - \alpha}{n_s^2} \sum_{i,j} \mathbf{S^s}_{i,j} \|T(\mathbf{x}^s_i) - T(\mathbf{x}^s_j) \|^2 + \frac{\alpha}{n_t^2} \sum_{i,j} \mathbf{S^t}_{i,j} \|T(\mathbf{x}^t_i) - T(\mathbf{x}^t_j) \|^2\]

with \(\mathbf{S^s}_{i,j}, \mathbf{S^t}_{i,j}\) denoting source and target similarity matrices and \(T(\cdot)\) being a barycentric mapping.

The algorithm used for solving the problem is the conditional gradient algorithm as proposed in [5].

Parameters

a (array-like (ns,)) – samples weights in the source domain
b (array-like (nt,)) – samples weights in the target domain
xs (array-like (ns,d)) – samples in the source domain
xt (array-like (nt,d)) – samples in the target domain
M (array-like (ns,nt)) – loss matrix
sim (string, optional) – Type of similarity (‘knn’ or ‘gauss’) used to construct the Laplacian.
sim_param (int or float, optional) – Parameter (number of the nearest neighbors for sim=’knn’ or bandwidth for sim=’gauss’) used to compute the Laplacian.
reg (string) – Type of Laplacian regularization
eta (float) – Regularization term for Laplacian regularization
alpha (float) – Regularization term for source domain’s importance in regularization
numItermax (int, optional) – Max number of iterations
stopThr (float, optional) – Stop threshold on error (inner emd solver) (>0)
numInnerItermax (int, optional) – Max number of iterations (inner CG solver)
stopInnerThr (float, optional) – Stop threshold on error (inner CG solver) (>0)
verbose (bool, optional) – Print information along iterations
log (bool, optional) – record log if True

Returns

gamma ((ns, nt) array-like) – Optimal transportation matrix for the given parameters
log (dict) – log dictionary return only if log==True in parameters

References

5: N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy, “Optimal Transport for Domain Adaptation,” in IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.PP, no.99, pp.1-1
30: R. Flamary, N. Courty, D. Tuia, A. Rakotomamonjy, “Optimal transport with Laplacian regularization: Applications to domain adaptation and shape matching,” in NIPS Workshop on Optimal Transport and Machine Learning OTML, 2014.

See also

ot.lp.emd: Unregularized OT
ot.optim.cg: General regularized OT

ot.da.joint_OT_mapping_kernel(xs, xt, mu=1, eta=0.001, kerneltype='gaussian', sigma=1, bias=False, verbose=False, verbose2=False, numItermax=100, numInnerItermax=10, stopInnerThr=1e-06, stopThr=1e-05, log=False, **kwargs)[source]

Joint OT and nonlinear mapping estimation with kernels as proposed in [8].

The function solves the following optimization problem:

\[ \begin{align}\begin{aligned}\min_{\gamma, L\in\mathcal{H}}\quad \|L(\mathbf{X_s}) - n_s\gamma \mathbf{X_t}\|^2_F + \mu \langle \gamma, \mathbf{M} \rangle_F + \eta \|L\|^2_\mathcal{H}\\s.t. \ \gamma \mathbf{1} = \mathbf{a}\\ \gamma^T \mathbf{1} = \mathbf{b}\\ \gamma \geq 0\end{aligned}\end{align} \]

where :

\(\mathbf{M}\) is the (ns, nt) squared euclidean cost matrix between samples in \(\mathbf{X_s}\) and \(\mathbf{X_t}\) (scaled by \(n_s\))
\(L\) is a \(n_s \times d\) linear operator on a kernel matrix that approximates the barycentric mapping
\(\mathbf{a}\) and \(\mathbf{b}\) are uniform source and target weights

The problem consist in solving jointly an optimal transport matrix \(\gamma\) and the nonlinear mapping that fits the barycentric mapping \(n_s\gamma \mathbf{X_t}\).

One can also estimate a mapping with constant bias (see supplementary material of [8]) using the bias optional argument.

The algorithm used for solving the problem is the block coordinate descent that alternates between updates of \(\mathbf{G}\) (using conditionnal gradient) and the update of \(\mathbf{L}\) using a classical kernel least square solver.

Parameters

xs (array-like (ns,d)) – samples in the source domain
xt (array-like (nt,d)) – samples in the target domain
mu (float,optional) – Weight for the linear OT loss (>0)
eta (float, optional) – Regularization term for the linear mapping L (>0)
kerneltype (str,optional) – kernel used by calling function ot.utils.kernel() (gaussian by default)
sigma (float, optional) – Gaussian kernel bandwidth.
bias (bool,optional) – Estimate linear mapping with constant bias
verbose (bool, optional) – Print information along iterations
verbose2 (bool, optional) – Print information along iterations
numItermax (int, optional) – Max number of BCD iterations
numInnerItermax (int, optional) – Max number of iterations (inner CG solver)
stopInnerThr (float, optional) – Stop threshold on error (inner CG solver) (>0)
stopThr (float, optional) – Stop threshold on relative loss decrease (>0)
log (bool, optional) – record log if True

Returns

gamma ((ns, nt) array-like) – Optimal transportation matrix for the given parameters
L ((ns, d) array-like) – Nonlinear mapping matrix ((\(n_s+1\), d) if bias)
log (dict) – log dictionary return only if log==True in parameters

References

8: M. Perrot, N. Courty, R. Flamary, A. Habrard, “Mapping estimation for discrete optimal transport”, Neural Information Processing Systems (NIPS), 2016.

See also

ot.lp.emd: Unregularized OT
ot.optim.cg: General regularized OT

ot.da.joint_OT_mapping_linear(xs, xt, mu=1, eta=0.001, bias=False, verbose=False, verbose2=False, numItermax=100, numInnerItermax=10, stopInnerThr=1e-06, stopThr=1e-05, log=False, **kwargs)[source]

Joint OT and linear mapping estimation as proposed in [8].

The function solves the following optimization problem:

\[ \begin{align}\begin{aligned}\min_{\gamma,L}\quad \|L(\mathbf{X_s}) - n_s\gamma \mathbf{X_t} \|^2_F + \mu \langle \gamma, \mathbf{M} \rangle_F + \eta \|L - \mathbf{I}\|^2_F\\s.t. \ \gamma \mathbf{1} = \mathbf{a}\\ \gamma^T \mathbf{1} = \mathbf{b}\\ \gamma \geq 0\end{aligned}\end{align} \]

where :

\(\mathbf{M}\) is the (ns, nt) squared euclidean cost matrix between samples in \(\mathbf{X_s}\) and \(\mathbf{X_t}\) (scaled by \(n_s\))
\(L\) is a \(d\times d\) linear operator that approximates the barycentric mapping
\(\mathbf{I}\) is the identity matrix (neutral linear mapping)
\(\mathbf{a}\) and \(\mathbf{b}\) are uniform source and target weights

The problem consist in solving jointly an optimal transport matrix \(\gamma\) and a linear mapping that fits the barycentric mapping \(n_s\gamma \mathbf{X_t}\).

One can also estimate a mapping with constant bias (see supplementary material of [8]) using the bias optional argument.

The algorithm used for solving the problem is the block coordinate descent that alternates between updates of \(\mathbf{G}\) (using conditionnal gradient) and the update of \(\mathbf{L}\) using a classical least square solver.

Parameters

xs (array-like (ns,d)) – samples in the source domain
xt (array-like (nt,d)) – samples in the target domain
mu (float,optional) – Weight for the linear OT loss (>0)
eta (float, optional) – Regularization term for the linear mapping L (>0)
bias (bool,optional) – Estimate linear mapping with constant bias
numItermax (int, optional) – Max number of BCD iterations
stopThr (float, optional) – Stop threshold on relative loss decrease (>0)
numInnerItermax (int, optional) – Max number of iterations (inner CG solver)
stopInnerThr (float, optional) – Stop threshold on error (inner CG solver) (>0)
verbose (bool, optional) – Print information along iterations
log (bool, optional) – record log if True

Returns

gamma ((ns, nt) array-like) – Optimal transportation matrix for the given parameters
L ((d, d) array-like) – Linear mapping matrix ((\(d+1\), d) if bias)
log (dict) – log dictionary return only if log==True in parameters

References

8: M. Perrot, N. Courty, R. Flamary, A. Habrard, “Mapping estimation for discrete optimal transport”, Neural Information Processing Systems (NIPS), 2016.

See also

ot.lp.emd: Unregularized OT
ot.optim.cg: General regularized OT

ot.da.sinkhorn_l1l2_gl(a, labels_a, b, M, reg, eta=0.1, numItermax=10, numInnerItermax=200, stopInnerThr=1e-09, verbose=False, log=False)[source]

Solve the entropic regularization optimal transport problem with group lasso regularization

The function solves the following optimization problem:

\[ \begin{align}\begin{aligned}\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F + \mathrm{reg} \cdot \Omega_e(\gamma) + \eta \ \Omega_g(\gamma)\\s.t. \ \gamma \mathbf{1} = \mathbf{a}\\ \gamma^T \mathbf{1} = \mathbf{b}\\ \gamma \geq 0\end{aligned}\end{align} \]

where :

\(\mathbf{M}\) is the (ns, nt) metric cost matrix
\(\Omega_e\) is the entropic regularization term \(\Omega_e(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})\)
\(\Omega_g\) is the group lasso regulaization term \(\Omega_g(\gamma)=\sum_{i,c} \|\gamma_{i,\mathcal{I}_c}\|^2\) where \(\mathcal{I}_c\) are the index of samples from class c in the source domain.
\(\mathbf{a}\) and \(\mathbf{b}\) are source and target weights (sum to 1)

The algorithm used for solving the problem is the generalised conditional gradient as proposed in [5, 7].

Parameters

a (array-like (ns,)) – samples weights in the source domain
labels_a (array-like (ns,)) – labels of samples in the source domain
b (array-like (nt,)) – samples in the target domain
M (array-like (ns,nt)) – loss matrix
reg (float) – Regularization term for entropic regularization >0
eta (float, optional) – Regularization term for group lasso regularization >0
numItermax (int, optional) – Max number of iterations
numInnerItermax (int, optional) – Max number of iterations (inner sinkhorn solver)
stopInnerThr (float, optional) – Stop threshold on error (inner sinkhorn solver) (>0)
verbose (bool, optional) – Print information along iterations
log (bool, optional) – record log if True

Returns

gamma ((ns, nt) array-like) – Optimal transportation matrix for the given parameters
log (dict) – log dictionary return only if log==True in parameters

References

5: N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy, “Optimal Transport for Domain Adaptation,” in IEEE Transactions on Pattern Analysis and Machine Intelligence , vol.PP, no.99, pp.1-1
7: Rakotomamonjy, A., Flamary, R., & Courty, N. (2015). Generalized conditional gradient: analysis of convergence and applications. arXiv preprint arXiv:1510.06567.

See also

ot.optim.gcg: Generalized conditional gradient for OT problems

ot.da.sinkhorn_lpl1_mm(a, labels_a, b, M, reg, eta=0.1, numItermax=10, numInnerItermax=200, stopInnerThr=1e-09, verbose=False, log=False)[source]

Solve the entropic regularization optimal transport problem with nonconvex group lasso regularization

The function solves the following optimization problem:

\[ \begin{align}\begin{aligned}\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F + \mathrm{reg} \cdot \Omega_e(\gamma) + \eta \ \Omega_g(\gamma)\\s.t. \ \gamma \mathbf{1} = \mathbf{a}\\ \gamma^T \mathbf{1} = \mathbf{b}\\ \gamma \geq 0\end{aligned}\end{align} \]

where :

\(\mathbf{M}\) is the (ns, nt) metric cost matrix
\(\Omega_e\) is the entropic regularization term \(\Omega_e (\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})\)
\(\Omega_g\) is the group lasso regularization term \(\Omega_g(\gamma)=\sum_{i,c} \|\gamma_{i,\mathcal{I}_c}\|^{1/2}_1\) where \(\mathcal{I}_c\) are the index of samples from class c in the source domain.
\(\mathbf{a}\) and \(\mathbf{b}\) are source and target weights (sum to 1)

The algorithm used for solving the problem is the generalized conditional gradient as proposed in [5, 7].

Parameters

a (array-like (ns,)) – samples weights in the source domain
labels_a (array-like (ns,)) – labels of samples in the source domain
b (array-like (nt,)) – samples weights in the target domain
M (array-like (ns,nt)) – loss matrix
reg (float) – Regularization term for entropic regularization >0
eta (float, optional) – Regularization term for group lasso regularization >0
numItermax (int, optional) – Max number of iterations
numInnerItermax (int, optional) – Max number of iterations (inner sinkhorn solver)
stopInnerThr (float, optional) – Stop threshold on error (inner sinkhorn solver) (>0)
verbose (bool, optional) – Print information along iterations
log (bool, optional) – record log if True

Returns

gamma ((ns, nt) array-like) – Optimal transportation matrix for the given parameters
log (dict) – log dictionary return only if log==True in parameters

References

5: N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy, “Optimal Transport for Domain Adaptation,” in IEEE Transactions on Pattern Analysis and Machine Intelligence , vol.PP, no.99, pp.1-1
7: Rakotomamonjy, A., Flamary, R., & Courty, N. (2015). Generalized conditional gradient: analysis of convergence and applications. arXiv preprint arXiv:1510.06567.

See also

ot.lp.emd: Unregularized OT
ot.bregman.sinkhorn: Entropic regularized OT
ot.optim.cg: General regularized OT

Classes

class ot.da.BaseTransport[source]

Base class for OTDA objects

Note

All estimators should specify all the parameters that can be set at the class level in their __init__ as explicit keyword arguments (no *args or **kwargs).

The fit method should:

estimate a cost matrix and store it in a cost_ attribute
estimate a coupling matrix and store it in a coupling_ attribute
estimate distributions from source and target data and store them in mu_s and mu_t attributes
store Xs and Xt in attributes to be used later on in transform and inverse_transform methods

transform method should always get as input a Xs parameter

inverse_transform method should always get as input a Xt parameter

transform_labels method should always get as input a ys parameter

inverse_transform_labels method should always get as input a yt parameter

fit(Xs=None, ys=None, Xt=None, yt=None)[source]

Build a coupling matrix from source and target sets of samples \((\mathbf{X_s}, \mathbf{y_s})\) and \((\mathbf{X_t}, \mathbf{y_t})\)

Parameters

Xs (array-like, shape (n_source_samples, n_features)) – The training input samples.
ys (array-like, shape (n_source_samples,)) – The training class labels
Xt (array-like, shape (n_target_samples, n_features)) – The training input samples.
yt (array-like, shape (n_target_samples,)) –
The class labels. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label

Returns

self – Returns self.

Return type

object

fit_transform(Xs=None, ys=None, Xt=None, yt=None)[source]

Build a coupling matrix from source and target sets of samples \((\mathbf{X_s}, \mathbf{y_s})\) and \((\mathbf{X_t}, \mathbf{y_t})\) and transports source samples \(\mathbf{X_s}\) onto target ones \(\mathbf{X_t}\)

Parameters

Xs (array-like, shape (n_source_samples, n_features)) – The training input samples.
ys (array-like, shape (n_source_samples,)) – The class labels for training samples
Xt (array-like, shape (n_target_samples, n_features)) – The training input samples.
yt (array-like, shape (n_target_samples,)) –
The class labels. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label

Returns

transp_Xs – The source samples samples.

Return type

array-like, shape (n_source_samples, n_features)

inverse_transform(Xs=None, ys=None, Xt=None, yt=None, batch_size=128)[source]

Transports target samples \(\mathbf{X_t}\) onto source samples \(\mathbf{X_s}\)

Parameters

Xs (array-like, shape (n_source_samples, n_features)) – The source input samples.
ys (array-like, shape (n_source_samples,)) – The source class labels
Xt (array-like, shape (n_target_samples, n_features)) – The target input samples.
yt (array-like, shape (n_target_samples,)) –
The target class labels. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label
batch_size (int, optional (default=128)) – The batch size for out of sample inverse transform

Returns

transp_Xt – The transported target samples.

Return type

array-like, shape (n_source_samples, n_features)

inverse_transform_labels(yt=None)[source]

Propagate target labels \(\mathbf{y_t}\) to obtain estimated source labels \(\mathbf{y_s}\)

Parameters: yt (array-like, shape (n_target_samples,)) –
Returns: transp_ys – Estimated soft source labels.
Return type: array-like, shape (n_source_samples, nb_classes)

transform(Xs=None, ys=None, Xt=None, yt=None, batch_size=128)[source]

Transports source samples \(\mathbf{X_s}\) onto target ones \(\mathbf{X_t}\)

Parameters

Xs (array-like, shape (n_source_samples, n_features)) – The source input samples.
ys (array-like, shape (n_source_samples,)) – The class labels for source samples
Xt (array-like, shape (n_target_samples, n_features)) – The target input samples.
yt (array-like, shape (n_target_samples,)) –
The class labels for target. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label
batch_size (int, optional (default=128)) – The batch size for out of sample inverse transform

Returns

transp_Xs – The transport source samples.

Return type

array-like, shape (n_source_samples, n_features)

transform_labels(ys=None)[source]

Propagate source labels \(\mathbf{y_s}\) to obtain estimated target labels as in [27].

Parameters: ys (array-like, shape (n_source_samples,)) – The source class labels
Returns: transp_ys – Estimated soft target labels.
Return type: array-like, shape (n_target_samples, nb_classes)

References

27: Ievgen Redko, Nicolas Courty, Rémi Flamary, Devis Tuia “Optimal transport for multi-source domain adaptation under target shift”, International Conference on Artificial Intelligence and Statistics (AISTATS), 2019.

Examples using `ot.da.BaseTransport`

class ot.da.EMDLaplaceTransport(reg_type='pos', reg_lap=1.0, reg_src=1.0, metric='sqeuclidean', norm=None, similarity='knn', similarity_param=None, max_iter=100, tol=1e-09, max_inner_iter=100000, inner_tol=1e-09, log=False, verbose=False, distribution_estimation=<function distribution_estimation_uniform>, out_of_sample_map='ferradans')[source]

Domain Adaptation OT method based on Earth Mover’s Distance with Laplacian regularization

Parameters

reg_type (string optional (default='pos')) – Type of the regularization term: ‘pos’ and ‘disp’ for regularization term defined in [2] and [6], respectively.
reg_lap (float, optional (default=1)) – Laplacian regularization parameter
reg_src (float, optional (default=0.5)) – Source relative importance in regularization
metric (string, optional (default="sqeuclidean")) – The ground metric for the Wasserstein problem
norm (string, optional (default=None)) – If given, normalize the ground metric to avoid numerical errors that can occur with large metric values.
similarity (string, optional (default="knn")) – The similarity to use either knn or gaussian
similarity_param (int or float, optional (default=None)) – Parameter for the similarity: number of nearest neighbors or bandwidth if similarity=”knn” or “gaussian”, respectively. If None is provided, it is set to 3 or the average pairwise squared Euclidean distance, respectively.
max_iter (int, optional (default=100)) – Max number of BCD iterations
tol (float, optional (default=1e-5)) – Stop threshold on relative loss decrease (>0)
max_inner_iter (int, optional (default=10)) – Max number of iterations (inner CG solver)
inner_tol (float, optional (default=1e-6)) – Stop threshold on error (inner CG solver) (>0)
log (int, optional (default=False)) – Controls the logs of the optimization algorithm
distribution_estimation (callable, optional (defaults to the uniform)) – The kind of distribution estimation to employ
out_of_sample_map (string, optional (default="ferradans")) – The kind of out of sample mapping to apply to transport samples from a domain into another one. Currently the only possible option is “ferradans” which uses the method proposed in [6].

coupling_

The optimal coupling

Type: array-like, shape (n_source_samples, n_target_samples)

References

1: N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy, “Optimal Transport for Domain Adaptation,” in IEEE Transactions on Pattern Analysis and Machine Intelligence , vol.PP, no.99, pp.1-1
2: R. Flamary, N. Courty, D. Tuia, A. Rakotomamonjy, “Optimal transport with Laplacian regularization: Applications to domain adaptation and shape matching,” in NIPS Workshop on Optimal Transport and Machine Learning OTML, 2014.
6: Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014). Regularized discrete optimal transport. SIAM Journal on Imaging Sciences, 7(3), 1853-1882.

fit(Xs, ys=None, Xt=None, yt=None)[source]

Build a coupling matrix from source and target sets of samples \((\mathbf{X_s}, \mathbf{y_s})\) and \((\mathbf{X_t}, \mathbf{y_t})\)

Parameters

Xs (array-like, shape (n_source_samples, n_features)) – The training input samples.
ys (array-like, shape (n_source_samples,)) – The class labels
Xt (array-like, shape (n_target_samples, n_features)) – The training input samples.
yt (array-like, shape (n_target_samples,)) –
The class labels. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label

Returns

self – Returns self.

Return type

object

Examples using `ot.da.EMDLaplaceTransport`

class ot.da.EMDTransport(metric='sqeuclidean', norm=None, log=False, distribution_estimation=<function distribution_estimation_uniform>, out_of_sample_map='ferradans', limit_max=10, max_iter=100000)[source]

Domain Adaptation OT method based on Earth Mover’s Distance

Parameters

metric (string, optional (default="sqeuclidean")) – The ground metric for the Wasserstein problem
norm (string, optional (default=None)) – If given, normalize the ground metric to avoid numerical errors that can occur with large metric values.
log (int, optional (default=False)) – Controls the logs of the optimization algorithm
distribution_estimation (callable, optional (defaults to the uniform)) – The kind of distribution estimation to employ
out_of_sample_map (string, optional (default="ferradans")) – The kind of out of sample mapping to apply to transport samples from a domain into another one. Currently the only possible option is “ferradans” which uses the method proposed in [6].
limit_max (float, optional (default=10)) – Controls the semi supervised mode. Transport between labeled source and target samples of different classes will exhibit an infinite cost (10 times the maximum value of the cost matrix)
max_iter (int, optional (default=100000)) – The maximum number of iterations before stopping the optimization algorithm if it has not converged.

coupling_

The optimal coupling

Type: array-like, shape (n_source_samples, n_target_samples)

References

1: N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy, “Optimal Transport for Domain Adaptation,” in IEEE Transactions on Pattern Analysis and Machine Intelligence , vol.PP, no.99, pp.1-1
6: Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014). Regularized discrete optimal transport. SIAM Journal on Imaging Sciences, 7(3), 1853-1882.

fit(Xs, ys=None, Xt=None, yt=None)[source]

Build a coupling matrix from source and target sets of samples \((\mathbf{X_s}, \mathbf{y_s})\) and \((\mathbf{X_t}, \mathbf{y_t})\)

Parameters

Xs (array-like, shape (n_source_samples, n_features)) – The training input samples.
ys (array-like, shape (n_source_samples,)) – The class labels
Xt (array-like, shape (n_target_samples, n_features)) – The training input samples.
yt (array-like, shape (n_target_samples,)) –
The class labels. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label

Returns

self – Returns self.

Return type

object

Examples using `ot.da.EMDTransport`

class ot.da.JCPOTTransport(reg_e=0.1, max_iter=10, tol=1e-08, verbose=False, log=False, metric='sqeuclidean', out_of_sample_map='ferradans')[source]

Domain Adaptation OT method for multi-source target shift based on Wasserstein barycenter algorithm.

Parameters

reg_e (float, optional (default=1)) – Entropic regularization parameter
max_iter (int, float, optional (default=10)) – The minimum number of iteration before stopping the optimization algorithm if it has not converged
tol (float, optional (default=10e-9)) – Stop threshold on error (inner sinkhorn solver) (>0)
verbose (bool, optional (default=False)) – Controls the verbosity of the optimization algorithm
log (bool, optional (default=False)) – Controls the logs of the optimization algorithm
metric (string, optional (default="sqeuclidean")) – The ground metric for the Wasserstein problem
norm (string, optional (default=None)) – If given, normalize the ground metric to avoid numerical errors that can occur with large metric values.
distribution_estimation (callable, optional (defaults to the uniform)) – The kind of distribution estimation to employ
out_of_sample_map (string, optional (default="ferradans")) – The kind of out of sample mapping to apply to transport samples from a domain into another one. Currently the only possible option is “ferradans” which uses the method proposed in [6].

coupling_

A set of optimal couplings between each source domain and the target domain

Type: list of array-like objects, shape K x (n_source_samples, n_target_samples)

proportions_

Estimated class proportions in the target domain

Type: array-like, shape (n_classes,)

log_

The dictionary of log, empty dict if parameter log is not True

Type: dictionary

References

1: Ievgen Redko, Nicolas Courty, Rémi Flamary, Devis Tuia “Optimal transport for multi-source domain adaptation under target shift”, International Conference on Artificial Intelligence and Statistics (AISTATS), vol. 89, p.849-858, 2019.
6: Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014). Regularized discrete optimal transport. SIAM Journal on Imaging Sciences, 7(3), 1853-1882.

fit(Xs, ys=None, Xt=None, yt=None)[source]

Building coupling matrices from a list of source and target sets of samples \((\mathbf{X_s}, \mathbf{y_s})\) and \((\mathbf{X_t}, \mathbf{y_t})\)

Parameters

Xs (list of K array-like objects, shape K x (nk_source_samples, n_features)) – A list of the training input samples.
ys (list of K array-like objects, shape K x (nk_source_samples,)) – A list of the class labels
Xt (array-like, shape (n_target_samples, n_features)) – The training input samples.
yt (array-like, shape (n_target_samples,)) –
The class labels. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label

Returns

self – Returns self.

Return type

object

inverse_transform_labels(yt=None)[source]

Propagate target labels \(\mathbf{y_t}\) to obtain estimated source labels \(\mathbf{y_s}\)

Parameters: yt (array-like, shape (n_target_samples,)) – The target class labels
Returns: transp_ys – A list of estimated soft source labels
Return type: list of K array-like objects, shape K x (nk_source_samples, nb_classes)

transform(Xs=None, ys=None, Xt=None, yt=None, batch_size=128)[source]

Transports source samples \(\mathbf{X_s}\) onto target ones \(\mathbf{X_t}\)

Parameters

Xs (list of K array-like objects, shape K x (nk_source_samples, n_features)) – A list of the training input samples.
ys (list of K array-like objects, shape K x (nk_source_samples,)) – A list of the class labels
Xt (array-like, shape (n_target_samples, n_features)) – The training input samples.
yt (array-like, shape (n_target_samples,)) –
The class labels. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label
batch_size (int, optional (default=128)) – The batch size for out of sample inverse transform

transform_labels(ys=None)[source]

Propagate source labels \(\mathbf{y_s}\) to obtain target labels as in [27]

Parameters: ys (list of K array-like objects, shape K x (nk_source_samples,)) – A list of the class labels
Returns: yt – Estimated soft target labels.
Return type: array-like, shape (n_target_samples, nb_classes)

References

27: Ievgen Redko, Nicolas Courty, Rémi Flamary, Devis Tuia “Optimal transport for multi-source domain adaptation under target shift”, International Conference on Artificial Intelligence and Statistics (AISTATS), 2019.

Examples using `ot.da.JCPOTTransport`

class ot.da.LinearTransport(reg=1e-08, bias=True, log=False, distribution_estimation=<function distribution_estimation_uniform>)[source]

OT linear operator between empirical distributions

The function estimates the optimal linear operator that aligns the two empirical distributions. This is equivalent to estimating the closed form mapping between two Gaussian distributions \(\mathcal{N}(\mu_s,\Sigma_s)\) and \(\mathcal{N}(\mu_t,\Sigma_t)\) as proposed in [14] and discussed in remark 2.29 in [15].

The linear operator from source to target \(M\)

\[M(\mathbf{x})= \mathbf{A} \mathbf{x} + \mathbf{b}\]

where :

\[ \begin{align}\begin{aligned}\mathbf{A} &= \Sigma_s^{-1/2} \left(\Sigma_s^{1/2}\Sigma_t\Sigma_s^{1/2} \right)^{1/2} \Sigma_s^{-1/2}\\\mathbf{b} &= \mu_t - \mathbf{A} \mu_s\end{aligned}\end{align} \]

Parameters

reg (float,optional) – regularization added to the daigonals of covariances (>0)
bias (boolean, optional) – estimate bias \(\mathbf{b}\) else \(\mathbf{b} = 0\) (default:True)
log (bool, optional) – record log if True

References

14: Knott, M. and Smith, C. S. “On the optimal mapping of distributions”, Journal of Optimization Theory and Applications Vol 43, 1984
15: Peyré, G., & Cuturi, M. (2017). “Computational Optimal Transport”, 2018.

fit(Xs=None, ys=None, Xt=None, yt=None)[source]

Build a coupling matrix from source and target sets of samples \((\mathbf{X_s}, \mathbf{y_s})\) and \((\mathbf{X_t}, \mathbf{y_t})\)

Parameters

Xs (array-like, shape (n_source_samples, n_features)) – The training input samples.
ys (array-like, shape (n_source_samples,)) – The class labels
Xt (array-like, shape (n_target_samples, n_features)) – The training input samples.
yt (array-like, shape (n_target_samples,)) –
The class labels. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label

Returns

self – Returns self.

Return type

object

inverse_transform(Xs=None, ys=None, Xt=None, yt=None, batch_size=128)[source]

Transports target samples \(\mathbf{X_t}\) onto source samples \(\mathbf{X_s}\)

Parameters

Xs (array-like, shape (n_source_samples, n_features)) – The training input samples.
ys (array-like, shape (n_source_samples,)) – The class labels
Xt (array-like, shape (n_target_samples, n_features)) – The training input samples.
yt (array-like, shape (n_target_samples,)) –
The class labels. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label
batch_size (int, optional (default=128)) – The batch size for out of sample inverse transform

Returns

transp_Xt – The transported target samples.

Return type

array-like, shape (n_source_samples, n_features)

transform(Xs=None, ys=None, Xt=None, yt=None, batch_size=128)[source]

Transports source samples \(\mathbf{X_s}\) onto target ones \(\mathbf{X_t}\)

Parameters

Xs (array-like, shape (n_source_samples, n_features)) – The training input samples.
ys (array-like, shape (n_source_samples,)) – The class labels
Xt (array-like, shape (n_target_samples, n_features)) – The training input samples.
yt (array-like, shape (n_target_samples,)) –
The class labels. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label
batch_size (int, optional (default=128)) – The batch size for out of sample inverse transform

Returns

transp_Xs – The transport source samples.

Return type

array-like, shape (n_source_samples, n_features)

Examples using `ot.da.LinearTransport`

class ot.da.MappingTransport(mu=1, eta=0.001, bias=False, metric='sqeuclidean', norm=None, kernel='linear', sigma=1, max_iter=100, tol=1e-05, max_inner_iter=10, inner_tol=1e-06, log=False, verbose=False, verbose2=False)[source]

MappingTransport: DA methods that aims at jointly estimating a optimal transport coupling and the associated mapping

Parameters

mu (float, optional (default=1)) – Weight for the linear OT loss (>0)
eta (float, optional (default=0.001)) – Regularization term for the linear mapping L (>0)
bias (bool, optional (default=False)) – Estimate linear mapping with constant bias
metric (string, optional (default="sqeuclidean")) – The ground metric for the Wasserstein problem
norm (string, optional (default=None)) – If given, normalize the ground metric to avoid numerical errors that can occur with large metric values.
kernel (string, optional (default="linear")) – The kernel to use either linear or gaussian
sigma (float, optional (default=1)) – The gaussian kernel parameter
max_iter (int, optional (default=100)) – Max number of BCD iterations
tol (float, optional (default=1e-5)) – Stop threshold on relative loss decrease (>0)
max_inner_iter (int, optional (default=10)) – Max number of iterations (inner CG solver)
inner_tol (float, optional (default=1e-6)) – Stop threshold on error (inner CG solver) (>0)
log (bool, optional (default=False)) – record log if True
verbose (bool, optional (default=False)) – Print information along iterations
verbose2 (bool, optional (default=False)) – Print information along iterations

coupling_

The optimal coupling

Type: array-like, shape (n_source_samples, n_target_samples)

mapping_

The associated mapping

array-like, shape (n_features (+ 1), n_features), (if bias) for kernel == linear
array-like, shape (n_source_samples (+ 1), n_features), (if bias) for kernel == gaussian

log_

The dictionary of log, empty dict if parameter log is not True

Type: dictionary

References

8: M. Perrot, N. Courty, R. Flamary, A. Habrard, “Mapping estimation for discrete optimal transport”, Neural Information Processing Systems (NIPS), 2016.

fit(Xs=None, ys=None, Xt=None, yt=None)[source]

Builds an optimal coupling and estimates the associated mapping from source and target sets of samples \((\mathbf{X_s}, \mathbf{y_s})\) and \((\mathbf{X_t}, \mathbf{y_t})\)

Parameters

Xs (array-like, shape (n_source_samples, n_features)) – The training input samples.
ys (array-like, shape (n_source_samples,)) – The class labels
Xt (array-like, shape (n_target_samples, n_features)) – The training input samples.
yt (array-like, shape (n_target_samples,)) –
The class labels. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label

Returns

self – Returns self

Return type

object

transform(Xs)[source]

Transports source samples \(\mathbf{X_s}\) onto target ones \(\mathbf{X_t}\)

Parameters: Xs (array-like, shape (n_source_samples, n_features)) – The training input samples.
Returns: transp_Xs – The transport source samples.
Return type: array-like, shape (n_source_samples, n_features)

Examples using `ot.da.MappingTransport`

class ot.da.SinkhornL1l2Transport(reg_e=1.0, reg_cl=0.1, max_iter=10, max_inner_iter=200, tol=1e-08, verbose=False, log=False, metric='sqeuclidean', norm=None, distribution_estimation=<function distribution_estimation_uniform>, out_of_sample_map='ferradans', limit_max=10)[source]

Domain Adaptation OT method based on sinkhorn algorithm + L1L2 class regularization.

Parameters

reg_e (float, optional (default=1)) – Entropic regularization parameter
reg_cl (float, optional (default=0.1)) – Class regularization parameter
max_iter (int, float, optional (default=10)) – The minimum number of iteration before stopping the optimization algorithm if it has not converged
max_inner_iter (int, float, optional (default=200)) – The number of iteration in the inner loop
tol (float, optional (default=10e-9)) – Stop threshold on error (inner sinkhorn solver) (>0)
verbose (bool, optional (default=False)) – Controls the verbosity of the optimization algorithm
log (bool, optional (default=False)) – Controls the logs of the optimization algorithm
metric (string, optional (default="sqeuclidean")) – The ground metric for the Wasserstein problem
norm (string, optional (default=None)) – If given, normalize the ground metric to avoid numerical errors that can occur with large metric values.
distribution_estimation (callable, optional (defaults to the uniform)) – The kind of distribution estimation to employ
out_of_sample_map (string, optional (default="ferradans")) – The kind of out of sample mapping to apply to transport samples from a domain into another one. Currently the only possible option is “ferradans” which uses the method proposed in [6].
limit_max (float, optional (default=10)) – Controls the semi supervised mode. Transport between labeled source and target samples of different classes will exhibit an infinite cost (10 times the maximum value of the cost matrix)

coupling_

The optimal coupling

Type: array-like, shape (n_source_samples, n_target_samples)

log_

The dictionary of log, empty dict if parameter log is not True

Type: dictionary

References

1: N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy, “Optimal Transport for Domain Adaptation,” in IEEE Transactions on Pattern Analysis and Machine Intelligence , vol.PP, no.99, pp.1-1
2: Rakotomamonjy, A., Flamary, R., & Courty, N. (2015). Generalized conditional gradient: analysis of convergence and applications. arXiv preprint arXiv:1510.06567.
6: Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014). Regularized discrete optimal transport. SIAM Journal on Imaging Sciences, 7(3), 1853-1882.

fit(Xs, ys=None, Xt=None, yt=None)[source]

Build a coupling matrix from source and target sets of samples \((\mathbf{X_s}, \mathbf{y_s})\) and \((\mathbf{X_t}, \mathbf{y_t})\)

Parameters

Xs (array-like, shape (n_source_samples, n_features)) – The training input samples.
ys (array-like, shape (n_source_samples,)) – The class labels
Xt (array-like, shape (n_target_samples, n_features)) – The training input samples.
yt (array-like, shape (n_target_samples,)) –
The class labels. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label

Returns

self – Returns self.

Return type

object

Examples using `ot.da.SinkhornL1l2Transport`

class ot.da.SinkhornLpl1Transport(reg_e=1.0, reg_cl=0.1, max_iter=10, max_inner_iter=200, log=False, tol=1e-08, verbose=False, metric='sqeuclidean', norm=None, distribution_estimation=<function distribution_estimation_uniform>, out_of_sample_map='ferradans', limit_max=inf)[source]

Domain Adaptation OT method based on sinkhorn algorithm + LpL1 class regularization.

Parameters

reg_e (float, optional (default=1)) – Entropic regularization parameter
reg_cl (float, optional (default=0.1)) – Class regularization parameter
max_iter (int, float, optional (default=10)) – The minimum number of iteration before stopping the optimization algorithm if it has not converged
max_inner_iter (int, float, optional (default=200)) – The number of iteration in the inner loop
log (bool, optional (default=False)) – Controls the logs of the optimization algorithm
tol (float, optional (default=10e-9)) – Stop threshold on error (inner sinkhorn solver) (>0)
verbose (bool, optional (default=False)) – Controls the verbosity of the optimization algorithm
metric (string, optional (default="sqeuclidean")) – The ground metric for the Wasserstein problem
norm (string, optional (default=None)) – If given, normalize the ground metric to avoid numerical errors that can occur with large metric values.
distribution_estimation (callable, optional (defaults to the uniform)) – The kind of distribution estimation to employ
out_of_sample_map (string, optional (default="ferradans")) – The kind of out of sample mapping to apply to transport samples from a domain into another one. Currently the only possible option is “ferradans” which uses the method proposed in [6].
limit_max (float, optional (default=np.infty)) – Controls the semi supervised mode. Transport between labeled source and target samples of different classes will exhibit a cost defined by limit_max.

coupling_

The optimal coupling

Type: array-like, shape (n_source_samples, n_target_samples)

References

1: N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy, “Optimal Transport for Domain Adaptation,” in IEEE Transactions on Pattern Analysis and Machine Intelligence , vol.PP, no.99, pp.1-1
2: Rakotomamonjy, A., Flamary, R., & Courty, N. (2015). Generalized conditional gradient: analysis of convergence and applications. arXiv preprint arXiv:1510.06567.
6: Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014). Regularized discrete optimal transport. SIAM Journal on Imaging Sciences, 7(3), 1853-1882.

fit(Xs, ys=None, Xt=None, yt=None)[source]

Build a coupling matrix from source and target sets of samples \((\mathbf{X_s}, \mathbf{y_s})\) and \((\mathbf{X_t}, \mathbf{y_t})\)

Parameters

Xs (array-like, shape (n_source_samples, n_features)) – The training input samples.
ys (array-like, shape (n_source_samples,)) – The class labels
Xt (array-like, shape (n_target_samples, n_features)) – The training input samples.
yt (array-like, shape (n_target_samples,)) –
The class labels. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label

Returns

self – Returns self.

Return type

object

Examples using `ot.da.SinkhornLpl1Transport`

class ot.da.SinkhornTransport(reg_e=1.0, max_iter=1000, tol=1e-08, verbose=False, log=False, metric='sqeuclidean', norm=None, distribution_estimation=<function distribution_estimation_uniform>, out_of_sample_map='ferradans', limit_max=inf)[source]

Domain Adaptation OT method based on Sinkhorn Algorithm

Parameters

reg_e (float, optional (default=1)) – Entropic regularization parameter
max_iter (int, float, optional (default=1000)) – The minimum number of iteration before stopping the optimization algorithm if it has not converged
tol (float, optional (default=10e-9)) – The precision required to stop the optimization algorithm.
verbose (bool, optional (default=False)) – Controls the verbosity of the optimization algorithm
log (int, optional (default=False)) – Controls the logs of the optimization algorithm
metric (string, optional (default="sqeuclidean")) – The ground metric for the Wasserstein problem
norm (string, optional (default=None)) – If given, normalize the ground metric to avoid numerical errors that can occur with large metric values.
distribution_estimation (callable, optional (defaults to the uniform)) – The kind of distribution estimation to employ
out_of_sample_map (string, optional (default="ferradans")) – The kind of out of sample mapping to apply to transport samples from a domain into another one. Currently the only possible option is “ferradans” which uses the method proposed in [6].
limit_max (float, optional (default=np.infty)) – Controls the semi supervised mode. Transport between labeled source and target samples of different classes will exhibit an cost defined by this variable

coupling_

The optimal coupling

Type: array-like, shape (n_source_samples, n_target_samples)

log_

The dictionary of log, empty dict if parameter log is not True

Type: dictionary

References

1: N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy, “Optimal Transport for Domain Adaptation,” in IEEE Transactions on Pattern Analysis and Machine Intelligence , vol.PP, no.99, pp.1-1
2: M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport, Advances in Neural Information Processing Systems (NIPS) 26, 2013
6: Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014). Regularized discrete optimal transport. SIAM Journal on Imaging Sciences, 7(3), 1853-1882.

fit(Xs=None, ys=None, Xt=None, yt=None)[source]

Build a coupling matrix from source and target sets of samples \((\mathbf{X_s}, \mathbf{y_s})\) and \((\mathbf{X_t}, \mathbf{y_t})\)

Parameters

Xs (array-like, shape (n_source_samples, n_features)) – The training input samples.
ys (array-like, shape (n_source_samples,)) – The class labels
Xt (array-like, shape (n_target_samples, n_features)) – The training input samples.
yt (array-like, shape (n_target_samples,)) –
The class labels. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label

Returns

self – Returns self.

Return type

object

Examples using `ot.da.SinkhornTransport`

class ot.da.UnbalancedSinkhornTransport(reg_e=1.0, reg_m=0.1, method='sinkhorn', max_iter=10, tol=1e-09, verbose=False, log=False, metric='sqeuclidean', norm=None, distribution_estimation=<function distribution_estimation_uniform>, out_of_sample_map='ferradans', limit_max=10)[source]

Domain Adaptation unbalanced OT method based on sinkhorn algorithm

Parameters

reg_e (float, optional (default=1)) – Entropic regularization parameter
reg_m (float, optional (default=0.1)) – Mass regularization parameter
method (str) – method used for the solver either ‘sinkhorn’, ‘sinkhorn_stabilized’ or ‘sinkhorn_epsilon_scaling’, see those function for specific parameters
max_iter (int, float, optional (default=10)) – The minimum number of iteration before stopping the optimization algorithm if it has not converged
tol (float, optional (default=10e-9)) – Stop threshold on error (inner sinkhorn solver) (>0)
verbose (bool, optional (default=False)) – Controls the verbosity of the optimization algorithm
log (bool, optional (default=False)) – Controls the logs of the optimization algorithm
metric (string, optional (default="sqeuclidean")) – The ground metric for the Wasserstein problem
norm (string, optional (default=None)) – If given, normalize the ground metric to avoid numerical errors that can occur with large metric values.
distribution_estimation (callable, optional (defaults to the uniform)) – The kind of distribution estimation to employ
out_of_sample_map (string, optional (default="ferradans")) – The kind of out of sample mapping to apply to transport samples from a domain into another one. Currently the only possible option is “ferradans” which uses the method proposed in [6].
limit_max (float, optional (default=10)) – Controls the semi supervised mode. Transport between labeled source and target samples of different classes will exhibit an infinite cost (10 times the maximum value of the cost matrix)

coupling_

The optimal coupling

Type: array-like, shape (n_source_samples, n_target_samples)

log_

The dictionary of log, empty dict if parameter log is not True

Type: dictionary

References

1: Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.
6: Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014). Regularized discrete optimal transport. SIAM Journal on Imaging Sciences, 7(3), 1853-1882.

fit(Xs, ys=None, Xt=None, yt=None)[source]

Build a coupling matrix from source and target sets of samples \((\mathbf{X_s}, \mathbf{y_s})\) and \((\mathbf{X_t}, \mathbf{y_t})\)

Parameters

Xs (array-like, shape (n_source_samples, n_features)) – The training input samples.
ys (array-like, shape (n_source_samples,)) – The class labels
Xt (array-like, shape (n_target_samples, n_features)) – The training input samples.
yt (array-like, shape (n_target_samples,)) –
The class labels. If some target samples are unlabelled, fill the \(\mathbf{y_t}\)’s elements with -1.

Warning: Note that, due to this convention -1 cannot be used as a class label

Returns

self – Returns self.

Return type

object

ot.da

Functions

Examples using ot.da.OT_mapping_linear

Classes

Examples using ot.da.BaseTransport

Examples using ot.da.EMDLaplaceTransport

Examples using ot.da.EMDTransport

Examples using ot.da.JCPOTTransport

Examples using ot.da.LinearTransport

Examples using ot.da.MappingTransport

Examples using ot.da.SinkhornL1l2Transport

Examples using ot.da.SinkhornLpl1Transport

Examples using ot.da.SinkhornTransport

Examples using `ot.da.OT_mapping_linear`

Examples using `ot.da.BaseTransport`

Examples using `ot.da.EMDLaplaceTransport`

Examples using `ot.da.EMDTransport`

Examples using `ot.da.JCPOTTransport`

Examples using `ot.da.LinearTransport`

Examples using `ot.da.MappingTransport`

Examples using `ot.da.SinkhornL1l2Transport`

Examples using `ot.da.SinkhornLpl1Transport`

Examples using `ot.da.SinkhornTransport`