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<!DOCTYPE html>

<head>

<title>ot.da — POT Python Optimal Transport 0.9.8.dev documentation</title>

</head>

POT Python Optimal Transport

</a>

</form>

</div>

</div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu">

<ul>

<li class="toctree-l1"><a class="reference internal" href="../../index.html">POT: Python Optimal Transport</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../auto_examples/plot_quickstart_guide.html">Quickstart Guide</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../auto_examples/index.html">Examples gallery</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../user_guide.html">User guide</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../all.html">API and modules</a></li>

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<li class="toctree-l1"><a class="reference internal" href="../../contributors.html">Contributors</a></li>

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<a href="../../index.html">POT Python Optimal Transport</a>

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<li class="breadcrumb-item"><a href="../index.html">Module code</a></li>

</li>

</ul>

<hr/>

</div>

<h1>Source code for ot.da</h1><div class="highlight"><pre>

# -*- coding: utf-8 -*-

"""

Domain adaptation with optimal transport

"""

# Author: Remi Flamary <remi.flamary@unice.fr>

# Nicolas Courty <ncourty@irisa.fr>

# Michael Perrot <michael.perrot@univ-st-etienne.fr>

# Nathalie Gayraud <nat.gayraud@gmail.com>

# Ievgen Redko <ievgen.redko@univ-st-etienne.fr>

# Eloi Tanguy <eloi.tanguy@math.cnrs.fr>

#

# License: MIT License

import numpy as np

import warnings

from .backend import get_backend

from .bregman import sinkhorn, jcpot_barycenter

from .lp import emd

from .utils import (

unif,

dist,

kernel,

cost_normalization,

label_normalization,

laplacian,

dots,

)

from .utils import (

BaseEstimator,

check_params,

deprecated,

labels_to_masks,

list_to_array,

)

from .unbalanced import sinkhorn_unbalanced

from .gaussian import (

empirical_bures_wasserstein_mapping,

empirical_gaussian_gromov_wasserstein_mapping,

)

from .optim import cg

from .optim import gcg

from .mapping import (

nearest_brenier_potential_fit,

nearest_brenier_potential_predict_bounds,

joint_OT_mapping_linear,

joint_OT_mapping_kernel,

)

# value used in ys/yt to mark a sample whose label is unknown

MISSING_LABEL = -1

def sinkhorn_lpl1_mm(

a,

labels_a,

b,

M,

reg,

eta=0.1,

numItermax=10,

numInnerItermax=200,

stopInnerThr=1e-9,

verbose=False,

log=False,

):

r"""

Solve the entropic regularization optimal transport problem with non-convex

group lasso regularization

The function solves the following optimization problem:

.. math::

\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

where :

- :math:`\mathbf{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 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.

- :math:`\mathbf{a}` and :math:`\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 :ref:`[5, 7] <references-sinkhorn-lpl1-mm>`.

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-sinkhorn-lpl1-mm:

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

"""

a, labels_a, b, M = list_to_array(a, labels_a, b, M)

nx = get_backend(a, labels_a, b, M)

p = 0.5

epsilon = 1e-3

labels_u, labels_idx = nx.unique(labels_a, return_inverse=True)

n_labels = labels_u.shape[0]

unroll_labels_idx = nx.eye(n_labels, type_as=M)[labels_idx]

W = nx.zeros(M.shape, type_as=M)

for _ in range(numItermax):

Mreg = M + eta * W

if log:

transp, log = sinkhorn(

a,

b,

Mreg,

reg,

numItermax=numInnerItermax,

stopThr=stopInnerThr,

log=True,

)

else:

transp = sinkhorn(

a, b, Mreg, reg, numItermax=numInnerItermax, stopThr=stopInnerThr

)

# the transport has been computed

# check if classes are really separated

W = (

nx.repeat(transp.T[:, :, None], n_labels, axis=2)

* unroll_labels_idx[None, :, :]

)

W = nx.sum(W, axis=1)

W = nx.dot(W, unroll_labels_idx.T)

W = p * ((W.T + epsilon) ** (p - 1))

if log:

return transp, log

else:

return transp</div>

def sinkhorn_l1l2_gl(

a,

labels_a,

b,

M,

reg,

eta=0.1,

numItermax=10,

numInnerItermax=200,

stopInnerThr=1e-9,

eps=1e-12,

verbose=False,

log=False,

):

r"""

Solve the entropic regularization optimal transport problem with group

lasso regularization

The function solves the following optimization problem:

.. math::

\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

where :

- :math:`\mathbf{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 regularization term

:math:`\Omega_g(\gamma)=\sum_{i,c} \|\gamma_{i,\mathcal{I}_c}\|^2`

where :math:`\mathcal{I}_c` are the index of samples from class

`c` in the source domain.

- :math:`\mathbf{a}` and :math:`\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 :ref:`[5, 7] <references-sinkhorn-l1l2-gl>`.

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)

eps: float, optional (default=1e-12)

Small value to avoid division by zero

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-sinkhorn-l1l2-gl:

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

"""

unroll_labels_idx = nx.eye(n_labels, type_as=labels_u)[None, labels_idx]

def f(G):

G_split = nx.repeat(G.T[:, :, None], n_labels, axis=2)

return nx.sum(nx.norm(G_split * unroll_labels_idx, axis=1))

def df(G):

W = nx.norm(G_split * unroll_labels_idx, axis=1, keepdims=True)

G_norm = G_split / nx.clip(W, eps, None)

return nx.sum(G_norm, axis=2).T

return gcg(

a,

b,

M,

reg,

eta,

f,

df,

G0=None,

numItermax=numItermax,

numInnerItermax=numInnerItermax,

stopThr=stopInnerThr,

verbose=verbose,

log=log,

)</div>

OT_mapping_linear = deprecated(empirical_bures_wasserstein_mapping)

def emd_laplace(

a,

b,

xs,

xt,

M,

sim="knn",

sim_param=None,

reg="pos",

eta=1,

alpha=0.5,

numItermax=100,

stopThr=1e-9,

numInnerItermax=100000,

stopInnerThr=1e-9,

log=False,

verbose=False,

):

r"""Solve the optimal transport problem (OT) with Laplacian regularization

.. math::

\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

where:

- :math:`\mathbf{a}` and :math:`\mathbf{b}` are source and target weights (sum to 1)

- :math:`\mathbf{x_s}` and :math:`\mathbf{x_t}` are source and target samples

- :math:`\mathbf{M}` is the (`ns`, `nt`) metric cost matrix

- :math:`\Omega_\alpha` is the Laplacian regularization term

.. math::

\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 :math:`\mathbf{S^s}_{i,j}, \mathbf{S^t}_{i,j}` denoting source and target similarity

matrices and :math:`T(\cdot)` being a barycentric mapping.

The algorithm used for solving the problem is the conditional gradient algorithm as proposed in

:ref:`[5] <references-emd-laplace>`.

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-emd-laplace:

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

"""

if not isinstance(sim_param, (int, float, type(None))):

raise ValueError(

"Similarity parameter should be an int or a float. Got {type} instead.".format(

type=type(sim_param).__name__

)

a, b, xs, xt, M = list_to_array(a, b, xs, xt, M)

nx = get_backend(a, b, xs, xt, M)

if sim == "gauss":

if sim_param is None:

sim_param = 1 / (2 * (nx.mean(dist(xs, xs, "sqeuclidean")) ** 2))

sS = kernel(xs, xs, method=sim, sigma=sim_param)

sT = kernel(xt, xt, method=sim, sigma=sim_param)

elif sim == "knn":

if sim_param is None:

sim_param = 3

try:

from sklearn.neighbors import kneighbors_graph

except ImportError:

raise ValueError(

"scikit-learn must be installed to use knn similarity. Install with `$pip install scikit-learn`."

)

sS = nx.from_numpy(

kneighbors_graph(X=nx.to_numpy(xs), n_neighbors=int(sim_param)).toarray(),

type_as=xs,

)

sS = (sS + sS.T) / 2

sT = nx.from_numpy(

kneighbors_graph(X=nx.to_numpy(xt), n_neighbors=int(sim_param)).toarray(),

type_as=xt,

)

sT = (sT + sT.T) / 2

else:

raise ValueError(

'Unknown similarity type {sim}. Currently supported similarity types are "knn" and "gauss".'.format(

sim=sim

)

lS = laplacian(sS)

lT = laplacian(sT)

def f(G):

return alpha * nx.trace(dots(xt.T, G.T, lS, G, xt)) + (1 - alpha) * nx.trace(

dots(xs.T, G, lT, G.T, xs)

)

ls2 = lS + lS.T

lt2 = lT + lT.T

xt2 = nx.dot(xt, xt.T)

if reg == "disp":

Cs = -eta * alpha / xs.shape[0] * dots(ls2, xs, xt.T)

Ct = -eta * (1 - alpha) / xt.shape[0] * dots(xs, xt.T, lt2)

M = M + Cs + Ct

def df(G):

return alpha * dots(ls2, G, xt2) + (1 - alpha) * dots(xs, xs.T, G, lt2)

return cg(

a,

b,

M,

reg=eta,

f=f,

df=df,

G0=None,

numItermax=numItermax,

numItermaxEmd=numInnerItermax,

stopThr=stopThr,

stopThr2=stopInnerThr,

verbose=verbose,

log=log,

)</div>

def distribution_estimation_uniform(X):

r"""estimates a uniform distribution from an array of samples :math:`\mathbf{X}`

Parameters

----------

X : array-like, shape (n_samples, n_features)

The array of samples

Returns

-------

mu : array-like, shape (n_samples,)

The uniform distribution estimated from :math:`\mathbf{X}`

"""

return unif(X.shape[0], type_as=X)</div>

class BaseTransport(BaseEstimator):

"""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

"""

def fit(self, Xs=None, ys=None, Xt=None, yt=None):

r"""Build a coupling matrix from source and target sets of samples

:math:`(\mathbf{X_s}, \mathbf{y_s})` and :math:`(\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

:math:`\mathbf{y_t}`'s elements with -1.

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

class label

Returns

-------

self : object

Returns self.

"""

nx = self._get_backend(Xs, ys, Xt, yt)

# check the necessary inputs parameters are here

if check_params(Xs=Xs, Xt=Xt):

# pairwise distance

self.cost_ = dist(Xs, Xt, metric=self.metric)

self.cost_, self.norm_cost_ = cost_normalization(

self.cost_, self.norm, return_value=True

)

if (ys is not None) and (yt is not None):

if self.limit_max != np.inf:

self.limit_max = self.limit_max * nx.max(self.cost_)

# present_labels is a (ns, nt) matrix of {0, 1} such that

# the cell (i, j) is 1 iff both ys[i] and yt[j] are labeled

# (i.e. neither is masked with MISSING_LABEL)

present_ys = (ys != MISSING_LABEL) + nx.zeros(ys.shape, type_as=ys)

present_yt = (yt != MISSING_LABEL) + nx.zeros(yt.shape, type_as=yt)

present_labels = present_ys[:, None] @ present_yt[None, :]

# label_mismatch is a (ns, nt) matrix of {True, False} such that

# the cell (i, j) is True if ys[i] != yt[j]

label_mismatch = (ys[:, None] - yt[None, :]) != 0

# cost correction is a (ns, nt) matrix of {-Inf, float, Inf} such

# that he cells (i, j) has -Inf where there's no correction necessary

# by 'correction' we mean setting cost to a large value when two

# labeled samples have different labels

# we suppress potential RuntimeWarning caused by Inf multiplication

# (as we explicitly cover potential NANs later)

with warnings.catch_warnings():

warnings.simplefilter("ignore", category=RuntimeWarning)

cost_correction = label_mismatch * present_labels * self.limit_max

# this operation is necessary because 0 * Inf = NAN

# thus is irrelevant when limit_max is finite

cost_correction = nx.nan_to_num(cost_correction, -np.inf)

self.cost_ = nx.maximum(self.cost_, cost_correction)

# distribution estimation

self.mu_s = self.distribution_estimation(Xs)

self.mu_t = self.distribution_estimation(Xt)

# store arrays of samples

self.xs_ = Xs

self.xt_ = Xt

return self</div>

def fit_transform(self, Xs=None, ys=None, Xt=None, yt=None):

r"""Build a coupling matrix from source and target sets of samples

:math:`(\mathbf{X_s}, \mathbf{y_s})` and :math:`(\mathbf{X_t}, \mathbf{y_t})`

and transports source samples :math:`\mathbf{X_s}` onto target ones :math:`\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

:math:`\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 : array-like, shape (n_source_samples, n_features)

The source samples samples.

"""

return self.fit(Xs, ys, Xt, yt).transform(Xs, ys, Xt, yt)</div>

def transform(self, Xs=None, ys=None, Xt=None, yt=None, batch_size=128):

r"""Transports source samples :math:`\mathbf{X_s}` onto target ones :math:`\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

:math:`\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 : array-like, shape (n_source_samples, n_features)

The transport source samples.

"""

nx = self.nx

# check the necessary inputs parameters are here

if check_params(Xs=Xs):

if nx.array_equal(self.xs_, Xs):

# perform standard barycentric mapping

transp = self.coupling_ / nx.sum(self.coupling_, axis=1)[:, None]

# set nans to 0

transp = nx.nan_to_num(transp, nan=0, posinf=0, neginf=0)

# compute transported samples

transp_Xs = nx.dot(transp, self.xt_)

else:

# perform out of sample mapping

indices = nx.arange(Xs.shape[0])

batch_ind = [

indices[i : i + batch_size]

for i in range(0, len(indices), batch_size)

]

transp_Xs = []

for bi in batch_ind:

# get the nearest neighbor in the source domain

D0 = dist(Xs[bi], self.xs_)

idx = nx.argmin(D0, axis=1)

# transport the source samples

transp_Xs_ = nx.dot(transp, self.xt_)

# define the transported points

transp_Xs_ = transp_Xs_[idx, :] + Xs[bi] - self.xs_[idx, :]

transp_Xs.append(transp_Xs_)

transp_Xs = nx.concatenate(transp_Xs, axis=0)

return transp_Xs</div>

def transform_labels(self, ys=None):

r"""Propagate source labels :math:`\mathbf{y_s}` to obtain estimated target labels as in

:ref:`[27] <references-basetransport-transform-labels>`.

Parameters

----------

ys : array-like, shape (n_source_samples,)

The source class labels

Returns

-------

transp_ys : array-like, shape (n_target_samples, nb_classes)

Estimated soft target labels.

.. _references-basetransport-transform-labels:

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.

"""

nx = self.nx

# check the necessary inputs parameters are here

if check_params(ys=ys):

# perform label propagation

transp = self.coupling_ / nx.sum(self.coupling_, axis=0)[None, :]

# set nans to 0

# compute propagated labels

labels = label_normalization(ys)

masks = labels_to_masks(labels, nx=nx, type_as=transp)

transp_ys = nx.dot(masks.T, transp)

return transp_ys.T</div>

def inverse_transform(self, Xs=None, ys=None, Xt=None, yt=None, batch_size=128):

r"""Transports target samples :math:`\mathbf{X_t}` onto source samples :math:`\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

:math:`\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 : array-like, shape (n_source_samples, n_features)

The transported target samples.

"""

nx = self.nx

# check the necessary inputs parameters are here

if check_params(Xt=Xt):

if nx.array_equal(self.xt_, Xt):

# perform standard barycentric mapping

transp_ = self.coupling_.T / nx.sum(self.coupling_, 0)[:, None]

# set nans to 0

transp_ = nx.nan_to_num(transp_, nan=0, posinf=0, neginf=0)

# compute transported samples

transp_Xt = nx.dot(transp_, self.xs_)

else:

# perform out of sample mapping

indices = nx.arange(Xt.shape[0])

batch_ind = [

]

transp_Xt = []

for bi in batch_ind:

D0 = dist(Xt[bi], self.xt_)

# transport the target samples

transp_Xt_ = nx.dot(transp_, self.xs_)

# define the transported points

transp_Xt_ = transp_Xt_[idx, :] + Xt[bi] - self.xt_[idx, :]

transp_Xt.append(transp_Xt_)

transp_Xt = nx.concatenate(transp_Xt, axis=0)

return transp_Xt</div>

def inverse_transform_labels(self, yt=None):

r"""Propagate target labels :math:`\mathbf{y_t}` to obtain estimated source labels

:math:`\mathbf{y_s}`

Parameters

----------

yt : array-like, shape (n_target_samples,)

Returns

-------

transp_ys : array-like, shape (n_source_samples, nb_classes)

Estimated soft source labels.

"""

nx = self.nx

# check the necessary inputs parameters are here

if check_params(yt=yt):

# perform label propagation

transp = self.coupling_ / nx.sum(self.coupling_, 1)[:, None]

# set nans to 0

# compute propagated labels

labels = label_normalization(yt)

return transp_ys.T</div>

</div>

class LinearTransport(BaseTransport):

r"""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 :math:`\mathcal{N}(\mu_s,\Sigma_s)`

and :math:`\mathcal{N}(\mu_t,\Sigma_t)` as proposed in

:ref:`[14] <references-lineartransport>` and discussed in remark 2.29 in

:ref:`[15] <references-lineartransport>`.

The linear operator from source to target :math:`M`

.. math::

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

where :

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