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<!DOCTYPE html>

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

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POT Python Optimal Transport

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

"""

Weak optimal ransport solvers

"""

# Author: Remi Flamary <remi.flamary@polytehnique.edu>

#

# License: MIT License

from .backend import get_backend

from .optim import cg

import numpy as np

__all__ = ["weak_optimal_transport"]

def weak_optimal_transport(

Xa, Xb, a=None, b=None, verbose=False, log=False, G0=None, **kwargs

):

r"""Solves the weak optimal transport problem between two empirical distributions

.. math::

\gamma = \mathop{\arg \min}_\gamma \quad \sum_i \mathbf{a}_i \left(\mathbf{X^a}_i - \frac{1}{\mathbf{a}_i} \sum_j \gamma_{ij} \mathbf{X^b}_j \right)^2

s.t. \ \gamma \mathbf{1} = \mathbf{a}

\gamma^T \mathbf{1} = \mathbf{b}

\gamma \geq 0

where :

- :math:`X^a` and :math:`X^b` are the sample matrices.

- :math:`\mathbf{a}` and :math:`\mathbf{b}` are the sample weights

.. note:: This function is backend-compatible and will work on arrays

from all compatible backends. But the algorithm uses the C++ CPU backend

which can lead to copy overhead on GPU arrays.

Uses the conditional gradient algorithm to solve the problem proposed

in :ref:`[39] <references-weak>`.

Parameters

----------

Xa : (ns,d) array-like, float

Source samples

Xb : (nt,d) array-like, float

Target samples

a : (ns,) array-like, float

Source histogram (uniform weight if empty list)

b : (nt,) array-like, float

Target histogram (uniform weight if empty list))

G0 : (ns,nt) array-like, float

initial guess (default is indep joint density)

numItermax : int, optional

Max number of iterations

numItermaxEmd : int, optional

Max number of iterations for emd

stopThr : float, optional

Stop threshold on the relative variation (>0)

stopThr2 : float, optional

Stop threshold on the absolute variation (>0)

verbose : bool, optional

Print information along iterations

log : bool, optional

record log if True

Returns

-------

gamma: array-like, shape (ns, nt)

Optimal transportation matrix for the given

parameters

log: dict, optional

If input log is true, a dictionary containing the

cost and dual variables and exit status

.. _references-weak:

References

----------

.. [39] Gozlan, N., Roberto, C., Samson, P. M., & Tetali, P. (2017).

Kantorovich duality for general transport costs and applications.

Journal of Functional Analysis, 273(11), 3327-3405.

See Also

--------

ot.bregman.sinkhorn : Entropic regularized OT

ot.optim.cg : General regularized OT

"""

nx = get_backend(Xa, Xb)

Xa2 = nx.to_numpy(Xa)

Xb2 = nx.to_numpy(Xb)

if a is None:

a2 = np.ones((Xa.shape[0])) / Xa.shape[0]

else:

a2 = nx.to_numpy(a)

if b is None:

b2 = np.ones((Xb.shape[0])) / Xb.shape[0]

else:

b2 = nx.to_numpy(b)

# init uniform

if G0 is None:

T0 = a2[:, None] * b2[None, :]

else:

T0 = nx.to_numpy(G0)

# weak OT loss

def f(T):

return np.dot(a2, np.sum((Xa2 - np.dot(T, Xb2) / a2[:, None]) ** 2, 1))

# weak OT gradient

def df(T):

return -2 * np.dot(Xa2 - np.dot(T, Xb2) / a2[:, None], Xb2.T)

# solve with conditional gradient and return solution

if log:

res, log = cg(a2, b2, 0, 1, f, df, T0, log=log, verbose=verbose, **kwargs)

log["u"] = nx.from_numpy(log["u"], type_as=Xa)

log["v"] = nx.from_numpy(log["v"], type_as=Xb)

return nx.from_numpy(res, type_as=Xa), log

else:

return nx.from_numpy(

cg(a2, b2, 0, 1, f, df, T0, log=log, verbose=verbose, **kwargs), type_as=Xa

)</div>

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