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<!DOCTYPE html>

<head>

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

</head>

POT Python Optimal Transport

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

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

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

"""

Generic solvers for regularized OT or its semi-relaxed version.

"""

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

# Titouan Vayer <titouan.vayer@irisa.fr>

# Cédric Vincent-Cuaz <cedvincentcuaz@gmail.com>

# License: MIT License

import numpy as np

import warnings

from .lp import emd

from .bregman import sinkhorn

from .backend import get_backend

with warnings.catch_warnings():

warnings.simplefilter("ignore")

try:

from scipy.optimize._linesearch import scalar_search_armijo

except ModuleNotFoundError:

# scipy<1.8.0

from scipy.optimize.linesearch import scalar_search_armijo

# The corresponding scipy function does not work for matrices

def line_search_armijo(

f,

xk,

pk,

gfk,

old_fval,

args=(),

c1=1e-4,

alpha0=0.99,

alpha_min=0.0,

alpha_max=None,

nx=None,

**kwargs,

):

r"""

Armijo linesearch function that works with matrices

Find an approximate minimum of :math:`f(x_k + \alpha \cdot p_k)` that satisfies the

armijo conditions.

.. note:: If the loss function f returns a float (resp. a 1d array) then

the returned alpha and fa are float (resp. 1d arrays).

Parameters

----------

f : callable

loss function

xk : array-like

initial position

pk : array-like

descent direction

gfk : array-like

gradient of `f` at :math:`x_k`

old_fval : float or 1d array

loss value at :math:`x_k`

args : tuple, optional

arguments given to `f`

c1 : float, optional

:math:`c_1` const in armijo rule (>0)

alpha0 : float, optional

initial step (>0)

alpha_min : float, default=0.

minimum value for alpha

alpha_max : float, optional

maximum value for alpha

nx : backend, optional

If let to its default value None, a backend test will be conducted.

Returns

-------

alpha : float or 1d array

step that satisfy armijo conditions

fc : int

nb of function call

fa : float or 1d array

loss value at step alpha

"""

if nx is None:

xk0, pk0 = xk, pk

nx = get_backend(xk0, pk0)

else:

xk0, pk0 = xk, pk

if len(xk.shape) == 0:

xk = nx.reshape(xk, (-1,))

xk = nx.to_numpy(xk)

pk = nx.to_numpy(pk)

gfk = nx.to_numpy(gfk)

fc = [0]

def phi(alpha1):

# it's necessary to check boundary condition here for the coefficient

# as the callback could be evaluated for negative value of alpha by

# `scalar_search_armijo` function here:

#

# https://github.com/scipy/scipy/blob/11509c4a98edded6c59423ac44ca1b7f28fba1fd/scipy/optimize/linesearch.py#L686

#

# see more details https://github.com/PythonOT/POT/issues/502

alpha1 = np.clip(alpha1, alpha_min, alpha_max)

# The callable function operates on nx backend

fc[0] += 1

alpha10 = nx.from_numpy(alpha1)

fval = f(xk0 + alpha10 * pk0, *args)

if isinstance(fval, float):

# prevent bug from nx.to_numpy that can look for .cpu or .gpu

return fval

else:

return nx.to_numpy(fval)

if old_fval is None:

phi0 = phi(0.0)

elif isinstance(old_fval, float):

# prevent bug from nx.to_numpy that can look for .cpu or .gpu

phi0 = old_fval

else:

phi0 = nx.to_numpy(old_fval)

derphi0 = np.sum(pk * gfk) # Quickfix for matrices

alpha, phi1 = scalar_search_armijo(

phi, phi0, derphi0, c1=c1, alpha0=alpha0, amin=alpha_min

)

if alpha is None:

return 0.0, fc[0], nx.from_numpy(phi0, type_as=xk0)

else:

alpha = np.clip(alpha, alpha_min, alpha_max)

return (

nx.from_numpy(alpha, type_as=xk0),

fc[0],

nx.from_numpy(phi1, type_as=xk0),

)</div>

def generic_conditional_gradient(

a,

b,

M,

f,

df,

reg1,

reg2,

lp_solver,

line_search,

G0=None,

numItermax=200,

stopThr=1e-9,

stopThr2=1e-9,

verbose=False,

log=False,

nx=None,

**kwargs,

):

r"""

Solve the general regularized OT problem or its semi-relaxed version with

conditional gradient or generalized conditional gradient depending on the

provided linear program solver.

The function solves the following optimization problem if set as a conditional gradient:

.. math::

\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +

\mathrm{reg_1} \cdot f(\gamma)

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

\gamma^T \mathbf{1} &= \mathbf{b} \ (\text{optional constraint})

\gamma &\geq 0

where :

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

- :math:`f` is the regularization term (and `df` is its gradient)

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

The algorithm used for solving the problem is conditional gradient as discussed in :ref:`[1] <references-cg>`

The function solves the following optimization problem if set a generalized conditional gradient:

.. math::

\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +

\mathrm{reg_1}\cdot f(\gamma) + \mathrm{reg_2}\cdot\Omega(\gamma)

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

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

\gamma &\geq 0

where :

- :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`

The algorithm used for solving the problem is the generalized conditional gradient as discussed in :ref:`[5, 7] <references-gcg>`

Parameters

----------

a : array-like, shape (ns,)

samples weights in the source domain

b : array-like, shape (nt,)

samples weights in the target domain

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

loss matrix

f : function

Regularization function taking a transportation matrix as argument

df: function

Gradient of the regularization function taking a transportation matrix as argument

reg1 : float

Regularization term >0

reg2 : float,

Entropic Regularization term >0. Ignored if set to None.

lp_solver: function,

linear program solver for direction finding of the (generalized) conditional gradient.

This function must take the form `lp_solver(a, b, Mi, **kwargs)` with p:

`a` and `b` are sample weights in both domains; `Mi` is the gradient of

the regularized objective; optimal arguments via kwargs.

It must output an admissible transport plan.

For instance, for the general regularized OT problem with conditional gradient :ref:`[1] <references-cg>`:

def lp_solver(a, b, M, **kwargs):

return ot.emd(a, b, M)

or with the generalized conditional gradient instead :ref:`[5, 7] <references-gcg>`:

def lp_solver(a, b, Mi, **kwargs):

return ot.sinkhorn(a, b, Mi)

line_search: function,

Function to find the optimal step. This function must take the form

`line_search(cost, G, deltaG, Mi, cost_G, df_G, **kwargs)` with: `cost`

the cost function, `G` the transport plan, `deltaG` the conditional

gradient direction given by lp_solver, `Mi` the gradient of regularized

objective, `cost_G` the cost at G, `df_G` the gradient of the regularizer

at G. Two types of outputs are supported:

Instances such as `ot.optim.line_search_armijo` (generic solver),

`ot.gromov.solve_gromov_linesearch` (FGW problems),

`solve_semirelaxed_gromov_linesearch` (srFGW problems) and

`gcg_linesearch` (generalized cg), output : the line-search step alpha,

the number of iterations used in the solver if applicable and the loss

value at step alpha. These can be called e.g as:

def line_search(cost, G, deltaG, Mi, cost_G, df_G, **kwargs):

return ot.optim.line_search_armijo(cost, G, deltaG, Mi, cost_G, **kwargs)

Instances such as `ot.gromov.solve_partial_gromov_linesearch` for partial

(F)GW problems add as finale output, the next step gradient reading as

a convex combination of previously computed gradients, taking advantage of the regularizer

quadratic form.

G0 : array-like, shape (ns,nt), optional

initial guess (default is indep joint density)

numItermax : int, optional

Max number of iterations

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

nx : backend, optional

If let to its default value None, the backend will be deduced from other inputs.

**kwargs : dict

Parameters for linesearch

Returns

-------

gamma : (ns x nt) ndarray

Optimal transportation matrix for the given parameters

log : dict

log dictionary return only if log==True in parameters

.. _references-cg:

.. _references_gcg:

References

----------

.. [1] Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014).

Regularized discrete optimal transport. SIAM Journal on Imaging Sciences, 7(3), 1853-1882.

.. [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 optimal transport

ot.bregman.sinkhorn : Entropic regularized optimal transport

"""

if nx is None:

if isinstance(M, int) or isinstance(M, float):

nx = get_backend(a, b)

else:

loop = 1

if log:

log = {"loss": []}

if G0 is None:

G = nx.outer(a, b)

else:

# to not change G0 in place.

G = nx.copy(G0)

if reg2 is None:

def cost(G):

else:

def cost(G):

return nx.sum(M * G) + reg1 * f(G) + reg2 * nx.sum(G * nx.log(G))

cost_G = cost(G)

if log:

log["loss"].append(cost_G)

df_G = None

it = 0

if verbose:

print(

"{:5s}|{:12s}|{:8s}|{:8s}".format(

"It.", "Loss", "Relative loss", "Absolute loss"

)

+ "\n"

+ "-" * 48

)

print("{:5d}|{:8e}|{:8e}|{:8e}".format(it, cost_G, 0, 0))

while loop:

it += 1

old_cost_G = cost_G

# problem linearization

if df_G is None:

df_G = df(G)

Mi = M + reg1 * df_G

if reg2 is not None:

Mi = Mi + reg2 * (1 + nx.log(G))

# solve linear program

Gc, innerlog_ = lp_solver(a, b, Mi, **kwargs)

# line search

deltaG = Gc - G

res_line_search = line_search(cost, G, deltaG, Mi, cost_G, df_G, **kwargs)

if len(res_line_search) == 3:

# the line-search does not allow to update the gradient

alpha, fc, cost_G = res_line_search

df_G = None

else:

# the line-search allows to update the gradient directly

# e.g. while using quadratic losses as the gromov-wasserstein loss

alpha, fc, cost_G, df_G = res_line_search

G = G + alpha * deltaG

# test convergence

if it >= numItermax:

loop = 0

abs_delta_cost_G = abs(cost_G - old_cost_G)

relative_delta_cost_G = (

abs_delta_cost_G / abs(cost_G) if cost_G != 0.0 else np.nan

)

if relative_delta_cost_G < stopThr or abs_delta_cost_G < stopThr2:

loop = 0

if log:

if verbose:

if it % 20 == 0:

print(

)

+ "\n"

+ "-" * 48

)

print(

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

it, cost_G, relative_delta_cost_G, abs_delta_cost_G

)

if log:

log.update(innerlog_)

return G, log

else:

return G</div>

def cg(

a,

b,

M,

reg,

f,

df,

G0=None,

line_search=None,

numItermax=200,

numItermaxEmd=100000,

stopThr=1e-9,

stopThr2=1e-9,

verbose=False,

log=False,

nx=None,

**kwargs,

):

r"""

Solve the general regularized OT problem with conditional gradient

The function solves the following optimization problem:

.. math::

\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +

\mathrm{reg} \cdot f(\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:`f` is the regularization term (and `df` is its gradient)

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

The algorithm used for solving the problem is conditional gradient as discussed in :ref:`[1] <references-cg>`

Parameters

----------

a : array-like, shape (ns,)

samples weights in the source domain

b : array-like, shape (nt,)

samples in the target domain

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

loss matrix

reg : float

Regularization term >0

G0 : array-like, shape (ns,nt), optional

initial guess (default is indep joint density)

line_search: function,

Function to find the optimal step.

Default is None and calls a wrapper to line_search_armijo.

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

nx : backend, optional

If let to its default value None, the backend will be deduced from other inputs.

**kwargs : dict

Parameters for linesearch

Returns

-------

gamma : (ns x nt) ndarray

Optimal transportation matrix for the given parameters

log : dict

log dictionary return only if log==True in parameters

.. _references-cg:

References

----------

.. [1] Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014).

Regularized discrete optimal transport. SIAM Journal on Imaging Sciences, 7(3), 1853-1882.

See Also

--------

ot.lp.emd : Unregularized optimal transport

ot.bregman.sinkhorn : Entropic regularized optimal transport

"""

if nx is None:

else:

if line_search is None:

def line_search(cost, G, deltaG, Mi, cost_G, df_G, **kwargs):

return line_search_armijo(cost, G, deltaG, Mi, cost_G, nx=nx, **kwargs)

def lp_solver(a, b, M, **kwargs):

return emd(a, b, M, numItermaxEmd, log=True)

return generic_conditional_gradient(

a,

b,

M,

f,

df,

reg,

None,

lp_solver,

line_search,

G0=G0,

numItermax=numItermax,

stopThr=stopThr,

stopThr2=stopThr2,

verbose=verbose,

log=log,

nx=nx,

**kwargs,

)</div>

def semirelaxed_cg(

a,

b,

M,

reg,

f,

df,

G0=None,

line_search=None,

numItermax=200,

stopThr=1e-9,

stopThr2=1e-9,

verbose=False,

log=False,

nx=None,

**kwargs,

):

r"""

Solve the general regularized and semi-relaxed OT problem with conditional gradient

The function solves the following optimization problem:

.. math::

\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +

\mathrm{reg} \cdot f(\gamma)

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

\gamma &\geq 0

where :

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

- :math:`f` is the regularization term (and `df` is its gradient)

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

The algorithm used for solving the problem is conditional gradient as discussed in :ref:`[1] <references-cg>`

Parameters

----------

a : array-like, shape (ns,)

samples weights in the source domain

b : array-like, shape (nt,)

currently estimated samples weights in the target domain

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

loss matrix

reg : float

Regularization term >0

G0 : array-like, shape (ns,nt), optional

initial guess (default is indep joint density)

line_search: function,

Function to find the optimal step.

Default is None and calls a wrapper to line_search_armijo.

numItermax : int, optional

Max number of iterations

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

nx : backend, optional

If let to its default value None, the backend will be deduced from other inputs.

**kwargs : dict

Parameters for linesearch

Returns

-------

gamma : (ns x nt) ndarray

Optimal transportation matrix for the given parameters

log : dict

log dictionary return only if log==True in parameters

.. _references-cg:

References

----------

.. [48] Cédric Vincent-Cuaz, Rémi Flamary, Marco Corneli, Titouan Vayer, Nicolas Courty.

"Semi-relaxed Gromov-Wasserstein divergence and applications on graphs"

International Conference on Learning Representations (ICLR), 2021.

"""

if nx is None:

else:

if line_search is None:

def lp_solver(a, b, Mi, **kwargs):

# get minimum by rows as binary mask

min_ = nx.reshape(nx.min(Mi, axis=1), (-1, 1))

# instead of exact elements equal to min_ we consider a small margin (1e-15)

# for float precision issues. Then the mass is split uniformly

# between these elements.

Gc = nx.ones(1, type_as=a) * (Mi <= min_ + 1e-15)

Gc *= nx.reshape((a / nx.sum(Gc, axis=1)), (-1, 1))

# return by default an empty inner_log

return Gc, {}

return generic_conditional_gradient(

a,

b,

M,

f,

df,

reg,

None,

lp_solver,

line_search,

G0=G0,

numItermax=numItermax,

stopThr=stopThr,

stopThr2=stopThr2,

verbose=verbose,

log=log,

nx=nx,

**kwargs,

)</div>

def partial_cg(

a,

b,

a_extended,

b_extended,

M,

reg,

f,

df,

G0=None,

line_search=line_search_armijo,

numItermax=200,

stopThr=1e-9,

stopThr2=1e-9,

warn=True,

verbose=False,

log=False,

**kwargs,

):

r"""

Solve the general regularized partial OT problem with conditional gradient

The function solves the following optimization problem:

.. math::

\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +

\mathrm{reg} \cdot f(\gamma)

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

\gamma \mathbf{1} &= \mathbf{b}

\mathbf{1}^T \gamma^T \mathbf{1} = m &\leq \min\{\|\mathbf{p}\|_1, \|\mathbf{q}\|_1\}

\gamma &\geq 0

where :

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

- :math:`f` is the regularization term (and `df` is its gradient)

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

- `m` is the amount of mass to be transported

The algorithm used for solving the problem is conditional gradient as discussed in :ref:`[1] <references-cg>`

Parameters

----------

a : array-like, shape (ns,)

samples weights in the source domain

b : array-like, shape (nt,)

currently estimated samples weights in the target domain

a_extended : array-like, shape (ns + nb_dummies,)

samples weights in the source domain with added dummy nodes

b_extended : array-like, shape (nt + nb_dummies,)

currently estimated samples weights in the target domain with added dummy nodes

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

loss matrix

reg : float

Regularization term >0

G0 : array-like, shape (ns,nt), optional

initial guess (default is indep joint density)

line_search: function,

Function to find the optimal step.

Default is the armijo line-search.

numItermax : int, optional

Max number of iterations

stopThr : float, optional

Stop threshold on the relative variation (>0)

stopThr2 : float, optional

Stop threshold on the absolute variation (>0)

warn: bool, optional.

Whether to raise a warning when EMD did not converge.

verbose : bool, optional

Print information along iterations

log : bool, optional

record log if True

**kwargs : dict

Parameters for linesearch

Returns

-------

gamma : (ns x nt) ndarray

Optimal transportation matrix for the given parameters

log : dict

log dictionary return only if log==True in parameters

.. _references-partial-cg:

References

----------

.. [29] Chapel, L., Alaya, M., Gasso, G. (2020). "Partial Optimal

Transport with Applications on Positive-Unlabeled Learning".

NeurIPS.

"""

n, m = a.shape[0], b.shape[0]

n_extended, m_extended = a_extended.shape[0], b_extended.shape[0]

nb_dummies = n_extended - n

def lp_solver(a, b, Mi, **kwargs):

# add dummy nodes to Mi

Mi_extended = np.zeros((n_extended, m_extended), dtype=Mi.dtype)

Mi_extended[:n, :m] = Mi

Mi_extended[-nb_dummies:, -nb_dummies:] = np.max(M) * 1e2

G_extended, log_ = emd(

a_extended, b_extended, Mi_extended, numItermax, log=True

)

Gc = G_extended[:n, :m]

if warn:

if log_["warning"] is not None:

raise ValueError(

"Error in the EMD resolution: try to increase the"

" number of dummy points"

)

return Gc, log_

return generic_conditional_gradient(

a,

b,

M,

f,

df,

reg,

None,

lp_solver,

line_search,

G0=G0,

numItermax=numItermax,

stopThr=stopThr,

stopThr2=stopThr2,

verbose=verbose,

log=log,

**kwargs,

)</div>

def gcg(

a,

b,

M,

reg1,

reg2,

f,

df,

G0=None,

numItermax=10,

numInnerItermax=200,

stopThr=1e-9,

stopThr2=1e-9,

verbose=False,

log=False,

**kwargs,

):

r"""

Solve the general regularized OT problem with the generalized conditional gradient

The function solves the following optimization problem:

.. math::

\gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +

\mathrm{reg_1}\cdot\Omega(\gamma) + \mathrm{reg_2}\cdot f(\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` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`

- :math:`f` is the regularization term (and `df` is its gradient)

- :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 discussed in :ref:`[5, 7] <references-gcg>`

Parameters

----------

a : array-like, shape (ns,)

samples weights in the source domain

b : array-like, (nt,)

samples in the target domain

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

loss matrix

reg1 : float

Entropic Regularization term >0

reg2 : float

Second Regularization term >0

G0 : array-like, shape (ns, nt), optional

initial guess (default is indep joint density)

numItermax : int, optional

Max number of iterations

numInnerItermax : int, optional

Max number of iterations of Sinkhorn

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 : ndarray, shape (ns, nt)

Optimal transportation matrix for the given parameters

log : dict

log dictionary return only if log==True in parameters

.. _references-gcg:

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.cg : conditional gradient

"""

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