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<!DOCTYPE html>

<head>

<title>ot.mapping — POT Python Optimal Transport 0.9.6 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>

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

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

<li class="toctree-l1"><a class="reference internal" href="../../contributing.html">Contributing to POT</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../code_of_conduct.html">Code of conduct</a></li>

</ul>

</div>

</nav>

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

</nav>

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

<li class="breadcrumb-item active">ot.mapping</li>

</li>

</ul>

<hr/>

</div>

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

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

"""

Optimal Transport maps and variants

.. warning::

Note that by default the module is not imported in :mod:`ot`. In order to

use it you need to explicitly import :mod:`ot.mapping`

"""

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

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

#

# License: MIT License

from .backend import get_backend, to_numpy

from .lp import emd

import numpy as np

from .optim import cg

from .utils import dist, unif, list_to_array, kernel, dots

def nearest_brenier_potential_fit(

X,

V,

X_classes=None,

a=None,

b=None,

strongly_convex_constant=0.6,

gradient_lipschitz_constant=1.4,

its=100,

log=False,

init_method="barycentric",

solver=None,

):

r"""

Computes optimal values and gradients at X for a strongly convex potential :math:`\varphi` with Lipschitz gradients

on the partitions defined by `X_classes`, where :math:`\varphi` is optimal such that

:math:`\nabla \varphi \#\mu \approx \nu`, given samples :math:`X = x_1, \cdots, x_n \sim \mu` and

:math:`V = v_1, \cdots, v_n \sim \nu`. Finding such a potential that has the desired regularity on the

partition :math:`(E_k)_{k \in [K]}` (given by the classes `X_classes`) is equivalent to finding optimal values

`phi` for the :math:`\varphi(x_i)` and its gradients :math:`\nabla \varphi(x_i)` (variable`G`).

In practice, these optimal values are found by solving the following problem

.. math::

:nowrap:

\begin{gather*}

\text{min} \sum_{i,j}\pi_{i,j}\|g_i - v_j\|_2^2 \\

g_1,\cdots, g_n \in \mathbb{R}^d,\; \varphi_1, \cdots, \varphi_n \in \mathbb{R},\; \pi \in \Pi(a, b) \\

\text{s.t.}\ \forall k \in [K],\; \forall i,j \in I_k: \\

\varphi_i-\varphi_j-\langle g_j, x_i-x_j\rangle \geq c_1\|g_i - g_j\|_2^2

+ c_2\|x_i-x_j\|_2^2 - c_3\langle g_j-g_i, x_j -x_i \rangle.

\end{gather*}

The constants :math:`c_1, c_2, c_3` only depend on `strongly_convex_constant` and `gradient_lipschitz_constant`.

The constraint :math:`\pi \in \Pi(a, b)` denotes the fact that the matrix :math:`\pi` belong to the OT polytope

of marginals a and b. :math:`I_k` is the subset of :math:`[n]` of the i such that :math:`x_i` is in the

partition (or class) :math:`E_k`, i.e. `X_classes[i] == k`.

This problem is solved by alternating over the variable :math:`\pi` and the variables :math:`\varphi_i, g_i`.

For :math:`\pi`, the problem is the standard discrete OT problem, and for :math:`\varphi_i, g_i`, the

problem is a convex QCQP solved using :code:`cvxpy` (ECOS solver).

Accepts any compatible backend, but will perform the QCQP optimisation on Numpy arrays, and convert back at the end.

.. warning:: This function requires the CVXPY library

.. warning:: Accepts any backend but will convert to Numpy then back to the backend.

Parameters

----------

X : array-like (n, d)

reference points used to compute the optimal values phi and G

V : array-like (n, d)

values of the gradients at the reference points X

X_classes : array-like (n,), optional

classes of the reference points, defaults to a single class

a : array-like (n,), optional

weights for the reference points X, defaults to uniform

b : array-like (n,), optional

weights for the target points V, defaults to uniform

strongly_convex_constant : float, optional

constant for the strong convexity of the input potential phi, defaults to 0.6

gradient_lipschitz_constant : float, optional

constant for the Lipschitz property of the input gradient G, defaults to 1.4

its: int, optional

number of iterations, defaults to 100

log : bool, optional

record log if true

init_method : str, optional

'target' initialises G=V, 'barycentric' initialises at the image of X by

the barycentric projection

solver : str, optional

The CVXPY solver to use

Returns

-------

phi : array-like (n,)

optimal values of the potential at the points X

G : array-like (n, d)

optimal values of the gradients at the points X

log : dict, optional

If input log is true, a dictionary containing the values of the variables at each iteration, as well

as solver information

References

----------

.. [58] François-Pierre Paty, Alexandre d’Aspremont, and Marco Cuturi. Regularity as regularization:

Smooth and strongly convex brenier potentials in optimal transport. In International Conference

on Artificial Intelligence and Statistics, pages 1222–1232. PMLR, 2020.

See Also

--------

ot.mapping.nearest_brenier_potential_predict_bounds : Predicting SSNB images on new source data

ot.da.NearestBrenierPotential : BaseTransport wrapper for SSNB

"""

try:

import cvxpy as cvx

except ImportError:

print("Please install CVXPY to use this function")

return

assert (

X.shape == V.shape

), f"point shape should be the same as value shape, yet {X.shape} != {V.shape}"

nx = get_backend(X, V, X_classes, a, b)

X, V = to_numpy(X), to_numpy(V)

n, d = X.shape

if X_classes is not None:

X_classes = to_numpy(X_classes)

assert X_classes.size == n, "incorrect number of class items"

else:

X_classes = np.zeros(n)

a = unif(n) if a is None else nx.to_numpy(a)

b = unif(n) if b is None else nx.to_numpy(b)

assert a.shape[-1] == b.shape[-1] == n, "incorrect measure weight sizes"

assert init_method in [

"target",

"barycentric",

], f"Unsupported initialization method '{init_method}'"

if init_method == "target":

G_val = V

else: # Init G_val with barycentric projection

G_val = emd(a, b, dist(X, V)) @ V / a.reshape(n, 1)

phi_val = None

log_dict = {"G_list": [], "phi_list": [], "its": []}

for _ in range(its): # alternate optimisation iterations

cost_matrix = dist(G_val, V)

# optimise the plan

plan = emd(a, b, cost_matrix)

# optimise the values phi and the gradients G

phi = cvx.Variable(n)

G = cvx.Variable((n, d))

constraints = []

cost = 0

for i in range(n):

for j in range(n):

cost += cvx.sum_squares(G[i, :] - V[j, :]) * plan[i, j]

objective = cvx.Minimize(cost) # OT cost

c1, c2, c3 = _ssnb_qcqp_constants(

strongly_convex_constant, gradient_lipschitz_constant

)

for k in np.unique(X_classes): # constraints for the convex interpolation

for i in np.where(X_classes == k)[0]:

for j in np.where(X_classes == k)[0]:

constraints += [

phi[i]

>= phi[j]

+ G[j].T @ (X[i] - X[j])

+ c1 * cvx.sum_squares(G[i] - G[j])

+ c2 * cvx.sum_squares(X[i] - X[j])

- c3 * (G[j] - G[i]).T @ (X[j] - X[i])

]

problem = cvx.Problem(objective, constraints)

problem.solve(solver=solver)

phi_val, G_val = phi.value, G.value

it_log_dict = {

"solve_time": problem.solver_stats.solve_time,

"setup_time": problem.solver_stats.setup_time,

"num_iters": problem.solver_stats.num_iters,

"status": problem.status,

"value": problem.value,

}

if log:

log_dict["its"].append(it_log_dict)

log_dict["G_list"].append(G_val)

log_dict["phi_list"].append(phi_val)

# convert back to backend

phi_val = nx.from_numpy(phi_val)

G_val = nx.from_numpy(G_val)

if not log:

return phi_val, G_val

return phi_val, G_val, log_dict</div>

def _ssnb_qcqp_constants(strongly_convex_constant, gradient_lipschitz_constant):

r"""

Handy function computing the constants for the Nearest Brenier Potential QCQP problems

Parameters

----------

strongly_convex_constant : float

gradient_lipschitz_constant : float

Returns

-------

c1 : float

c2 : float

c3 : float

"""

assert (

0 < strongly_convex_constant < gradient_lipschitz_constant

), "incompatible regularity assumption"

c = 1 / (2 * (1 - strongly_convex_constant / gradient_lipschitz_constant))

c1 = c / gradient_lipschitz_constant

c2 = strongly_convex_constant * c

c3 = 2 * strongly_convex_constant * c / gradient_lipschitz_constant

return c1, c2, c3

def nearest_brenier_potential_predict_bounds(

X,

phi,

G,

Y,

X_classes=None,

Y_classes=None,

strongly_convex_constant=0.6,

gradient_lipschitz_constant=1.4,

log=False,

solver=None,

):

r"""

Compute the values of the lower and upper bounding potentials at the input points Y, using the potential optimal

values phi at X and their gradients G at X. The 'lower' potential corresponds to the method from :ref:`[58]`,

Equation 2, while the bounding property and 'upper' potential come from :ref:`[59]`, Theorem 3.14 (taking into

account the fact that this theorem's statement has a min instead of a max, which is a typo). Both potentials are

optimal for the SSNB problem.

If :math:`I_k` is the subset of :math:`[n]` of the i such that :math:`x_i` is in the partition (or class)

:math:`E_k`, for each :math:`y \in E_k`, this function solves the convex QCQP problems,

respectively for l: 'lower' and u: 'upper':

.. math::

:nowrap:

\begin{gather*}

(\varphi_{l}(x), \nabla \varphi_l(x)) = \text{argmin}\ t, \\

t\in \mathbb{R},\; g\in \mathbb{R}^d, \\

\text{s.t.} \forall j \in I_k,\; t-\varphi_j - \langle g_j, y-x_j \rangle \geq c_1\|g - g_j\|_2^2

+ c_2\|y-x_j\|_2^2 - c_3\langle g_j-g, x_j -y \rangle.

\end{gather*}

.. math::

:nowrap:

\begin{gather*}

(\varphi_{u}(x), \nabla \varphi_u(x)) = \text{argmax}\ t, \\

t\in \mathbb{R},\; g\in \mathbb{R}^d, \\

\text{s.t.} \forall i \in I_k,\; \varphi_i^* -t - \langle g, x_i-y \rangle \geq c_1\|g_i - g\|_2^2

+ c_2\|x_i-y\|_2^2 - c_3\langle g-g_i, y -x_i \rangle.

\end{gather*}

The constants :math:`c_1, c_2, c_3` only depend on `strongly_convex_constant` and `gradient_lipschitz_constant`.

.. warning:: This function requires the CVXPY library

.. warning:: Accepts any backend but will convert to Numpy then back to the backend.

Parameters

----------

X : array-like (n, d)

reference points used to compute the optimal values phi and G

X_classes : array-like (n,)

classes of the reference points

phi : array-like (n,)

optimal values of the potential at the points X

G : array-like (n, d)

optimal values of the gradients at the points X

Y : array-like (m, d)

input points

X_classes : array-like (n,), optional

classes of the reference points, defaults to a single class

Y_classes : array_like (m,), optional

classes of the input points, defaults to a single class

strongly_convex_constant : float, optional

constant for the strong convexity of the input potential phi, defaults to 0.6

gradient_lipschitz_constant : float, optional

constant for the Lipschitz property of the input gradient G, defaults to 1.4

log : bool, optional

record log if true

solver : str, optional

The CVXPY solver to use

Returns

-------

phi_lu: array-like (2, m)

values of the lower and upper bounding potentials at Y

G_lu: array-like (2, m, d)

gradients of the lower and upper bounding potentials at Y

log : dict, optional

If input log is true, a dictionary containing solver information

References

----------

.. [58] François-Pierre Paty, Alexandre d’Aspremont, and Marco Cuturi. Regularity as regularization:

Smooth and strongly convex brenier potentials in optimal transport. In International Conference

on Artificial Intelligence and Statistics, pages 1222–1232. PMLR, 2020.

.. [59] Adrien B Taylor. Convex interpolation and performance estimation of first-order methods for

convex optimization. PhD thesis, Catholic University of Louvain, Louvain-la-Neuve, Belgium,

2017.

See Also

--------

ot.mapping.nearest_brenier_potential_fit : Fitting the SSNB on source and target data

ot.da.NearestBrenierPotential : BaseTransport wrapper for SSNB

"""

try:

import cvxpy as cvx

except ImportError:

print("Please install CVXPY to use this function")

return

nx = get_backend(X, phi, G, Y)

X = to_numpy(X)

phi = to_numpy(phi)

G = to_numpy(G)

Y = to_numpy(Y)

m, d = Y.shape

if Y_classes is not None:

Y_classes = to_numpy(Y_classes)

assert Y_classes.size == m, "wrong number of class items for Y"

else:

Y_classes = np.zeros(m)

assert (

X.shape[1] == d

), f"incompatible dimensions between X: {X.shape} and Y: {Y.shape}"

n, _ = X.shape

if X_classes is not None:

X_classes = to_numpy(X_classes)

else:

strongly_convex_constant, gradient_lipschitz_constant

)

phi_lu = np.zeros((2, m))

G_lu = np.zeros((2, m, d))

log_dict = {}

for y_idx in range(m):

log_item = {}

# lower bound

phi_l_y = cvx.Variable(1)

G_l_y = cvx.Variable(d)

constraints = []

k = Y_classes[y_idx]

constraints += [

phi_l_y

>= phi[j]

+ G[j].T @ (Y[y_idx] - X[j])

+ c1 * cvx.sum_squares(G_l_y - G[j])

+ c2 * cvx.sum_squares(Y[y_idx] - X[j])

- c3 * (G[j] - G_l_y).T @ (X[j] - Y[y_idx])

]

phi_lu[0, y_idx] = phi_l_y.value

G_lu[0, y_idx] = G_l_y.value

if log:

log_item["l"] = {

"status": problem.status,

"value": problem.value,

}

# upper bound

phi_u_y = cvx.Variable(1)

G_u_y = cvx.Variable(d)

objective = cvx.Maximize(phi_u_y)

constraints = []

constraints += [

phi[i]

>= phi_u_y

+ G_u_y.T @ (X[i] - Y[y_idx])

+ c2 * cvx.sum_squares(X[i] - Y[y_idx])

- c3 * (G_u_y - G[i]).T @ (Y[y_idx] - X[i])

]

phi_lu[1, y_idx] = phi_u_y.value

G_lu[1, y_idx] = G_u_y.value

if log:

log_item["u"] = {

"status": problem.status,

"value": problem.value,

}

log_dict[y_idx] = log_item

phi_lu, G_lu = nx.from_numpy(phi_lu), nx.from_numpy(G_lu)

if not log:

return phi_lu, G_lu

return phi_lu, G_lu, log_dict</div>

def joint_OT_mapping_linear(

xs,

xt,

mu=1,

eta=0.001,

bias=False,

verbose=False,

verbose2=False,

numItermax=100,

numInnerItermax=10,

stopInnerThr=1e-6,

stopThr=1e-5,

log=False,

**kwargs,

):

r"""Joint OT and linear mapping estimation as proposed in

:ref:`[8] <references-joint-OT-mapping-linear>`.

The function solves the following optimization problem:

.. math::

\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

where :

- :math:`\mathbf{M}` is the (`ns`, `nt`) squared euclidean cost matrix between samples in

:math:`\mathbf{X_s}` and :math:`\mathbf{X_t}` (scaled by :math:`n_s`)

- :math:`L` is a :math:`d\times d` linear operator that approximates the barycentric

mapping

- :math:`\mathbf{I}` is the identity matrix (neutral linear mapping)

- :math:`\mathbf{a}` and :math:`\mathbf{b}` are uniform source and target weights

The problem consist in solving jointly an optimal transport matrix

:math:`\gamma` and a linear mapping that fits the barycentric mapping

:math:`n_s\gamma \mathbf{X_t}`.

One can also estimate a mapping with constant bias (see supplementary

material of :ref:`[8] <references-joint-OT-mapping-linear>`) using the bias optional argument.

The algorithm used for solving the problem is the block coordinate

descent that alternates between updates of :math:`\mathbf{G}` (using conditional gradient)

and the update of :math:`\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 ((:math:`d+1`, `d`) if bias)

log : dict

log dictionary return only if log==True in parameters

.. _references-joint-OT-mapping-linear:

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

"""

xs, xt = list_to_array(xs, xt)

nx = get_backend(xs, xt)

ns, nt, d = xs.shape[0], xt.shape[0], xt.shape[1]

if bias:

xs1 = nx.concatenate((xs, nx.ones((ns, 1), type_as=xs)), axis=1)

xstxs = nx.dot(xs1.T, xs1)

Id = nx.eye(d + 1, type_as=xs)

Id[-1] = 0

I0 = Id[:, :-1]

def sel(x):

return x[:-1, :]

else:

xs1 = xs

Id = nx.eye(d, type_as=xs)

I0 = Id

def sel(x):

return x

if log:

log = {"err": []}

a = unif(ns, type_as=xs)

b = unif(nt, type_as=xt)

M = dist(xs, xt) * ns

G = emd(a, b, M)

vloss = []

def loss(L, G):

"""Compute full loss"""

return (

nx.sum((nx.dot(xs1, L) - ns * nx.dot(G, xt)) ** 2)

+ mu * nx.sum(G * M)

+ eta * nx.sum(sel(L - I0) ** 2)

)

def solve_L(G):

"""solve L problem with fixed G (least square)"""

xst = ns * nx.dot(G, xt)

return nx.solve(xstxs + eta * Id, nx.dot(xs1.T, xst) + eta * I0)

def solve_G(L, G0):

"""Update G with CG algorithm"""

xsi = nx.dot(xs1, L)

def f(G):

return nx.sum((xsi - ns * nx.dot(G, xt)) ** 2)

def df(G):

return -2 * ns * nx.dot(xsi - ns * nx.dot(G, xt), xt.T)

G = cg(

a,

b,

M,

1.0 / mu,

f,

df,

G0=G0,

numItermax=numInnerItermax,

stopThr=stopInnerThr,

)

return G

L = solve_L(G)

vloss.append(loss(L, G))

if verbose:

print(

"{:5s}|{:12s}|{:8s}".format("It.", "Loss", "Delta loss") + "\n" + "-" * 32

)

print("{:5d}|{:8e}|{:8e}".format(0, vloss[-1], 0))

# init loop

if numItermax > 0:

loop = 1

else:

loop = 0

it = 0

while loop:

it += 1

# update G

G = solve_G(L, G)

# update L

L = solve_L(G)

if it >= numItermax:

loop = 0

if abs(vloss[-1] - vloss[-2]) / abs(vloss[-2]) < stopThr:

loop = 0

if verbose:

if it % 20 == 0:

print(

+ "\n"

+ "-" * 32

)

print(

"{:5d}|{:8e}|{:8e}".format(

it, vloss[-1], (vloss[-1] - vloss[-2]) / abs(vloss[-2])

)

if log:

log["loss"] = vloss

return G, L, log

else:

return G, L</div>

def 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-6,

stopThr=1e-5,

log=False,

**kwargs,

):

r"""Joint OT and nonlinear mapping estimation with kernels as proposed in

:ref:`[8] <references-joint-OT-mapping-kernel>`.

The function solves the following optimization problem:

.. math::

\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

where :

- :math:`\mathbf{M}` is the (`ns`, `nt`) squared euclidean cost matrix between samples in

:math:`\mathbf{X_s}` and :math:`\mathbf{X_t}` (scaled by :math:`n_s`)

- :math:`L` is a :math:`n_s \times d` linear operator on a kernel matrix that

approximates the barycentric mapping

- :math:`\mathbf{a}` and :math:`\mathbf{b}` are uniform source and target weights

The problem consist in solving jointly an optimal transport matrix

:math:`\gamma` and the nonlinear mapping that fits the barycentric mapping

:math:`n_s\gamma \mathbf{X_t}`.

One can also estimate a mapping with constant bias (see supplementary

material of :ref:`[8] <references-joint-OT-mapping-kernel>`) using the bias optional argument.

The algorithm used for solving the problem is the block coordinate

descent that alternates between updates of :math:`\mathbf{G}` (using conditional gradient)

and the update of :math:`\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 :py:func:`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 ((:math:`n_s+1`, `d`) if bias)

log : dict

log dictionary return only if log==True in parameters

.. _references-joint-OT-mapping-kernel:

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

"""

ns, nt = xs.shape[0], xt.shape[0]

K = kernel(xs, xs, method=kerneltype, sigma=sigma)

if bias:

K1 = nx.concatenate((K, nx.ones((ns, 1), type_as=xs)), axis=1)

Id = nx.eye(ns + 1, type_as=xs)

Id[-1] = 0

Kp = nx.eye(ns + 1, type_as=xs)

Kp[:ns, :ns] = K

# ls regu

# K0 = K1.T.dot(K1)+eta*I

# Kreg=I

# RKHS regul

K0 = nx.dot(K1.T, K1) + eta * Kp

Kreg = Kp

else:

K1 = K

Id = nx.eye(ns, type_as=xs)

# ls regul

# K0 = K1.T.dot(K1)+eta*I

# Kreg=I

# proper kernel ridge

K0 = K + eta * Id

Kreg = K

if log:

log = {"err": []}

vloss = []

"""Compute full loss"""

return (

nx.sum((nx.dot(K1, L) - ns * nx.dot(G, xt)) ** 2)

+ eta * nx.trace(dots(L.T, Kreg, L))

)

def solve_L_nobias(G):

"""solve L problem with fixed G (least square)"""

def solve_L_bias(G):

"""solve L problem with fixed G (least square)"""

return nx.solve(K0, nx.dot(K1.T, xst))

"""Update G with CG algorithm"""

xsi = nx.dot(K1, L)

def f(G):

def df(G):

G = cg(

a,

b,

M,

1.0 / mu,

f,

df,

G0=G0,

numItermax=numInnerItermax,

stopThr=stopInnerThr,

)

return G

if bias:

solve_L = solve_L_bias

else:

solve_L = solve_L_nobias

L = solve_L(G)

if verbose:

print(

)

# init loop

if numItermax > 0:

loop = 1

else:

loop = 0

it = 0

while loop:

it += 1

# update G

# update L

L = solve_L(G)

if it >= numItermax:

loop = 0

if verbose:

if it % 20 == 0:

print(

+ "\n"

+ "-" * 32

)

print(

)

if log:

log["loss"] = vloss

return G, L, log

else:

return G, L</div>

</pre></div>

</div>

<hr/>

</div>

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