"""
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"]
[docs]
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
)
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