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<!DOCTYPE html>

<head>

<title>ot.smooth — POT Python Optimal Transport 0.9.7 documentation</title>

</head>

POT Python Optimal Transport

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

#

# Redistribution and use in source and binary forms, with or without

# modification, are permitted provided that the following conditions are met:

#

# 1. Redistributions of source code must retain the above copyright notice, this

# list of conditions and the following disclaimer.

#

# 2. Redistributions in binary form must reproduce the above copyright notice,

# this list of conditions and the following disclaimer in the documentation and/or

# other materials provided with the distribution.

#

# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND

# ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED

# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.

# IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT,

# INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT

# NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA,

# OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF

# LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR

# OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF

# THE POSSIBILITY OF SUCH DAMAGE.

# Author: Mathieu Blondel

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

# Tianlin Liu <t.liu@unibas.ch>

"""

Smooth and Sparse (KL an L2 reg.) and sparsity-constrained OT solvers.

Implementation of :

Smooth and Sparse Optimal Transport.

Mathieu Blondel, Vivien Seguy, Antoine Rolet.

In Proc. of AISTATS 2018.

https://arxiv.org/abs/1710.06276

(Original code from https://github.com/mblondel/smooth-ot/)

Sparsity-Constrained Optimal Transport.

Liu, T., Puigcerver, J., & Blondel, M. (2023).

Sparsity-constrained optimal transport.

Proceedings of the Eleventh International Conference on

Learning Representations (ICLR).

https://arxiv.org/abs/2209.15466

[17] Blondel, M., Seguy, V., & Rolet, A. (2018). Smooth and Sparse Optimal

Transport. Proceedings of the Twenty-First International Conference on

Artificial Intelligence and Statistics (AISTATS).

[50] Liu, T., Puigcerver, J., & Blondel, M. (2023).

Sparsity-constrained optimal transport.

Proceedings of the Eleventh International Conference on

Learning Representations (ICLR).

"""

import numpy as np

from scipy.optimize import minimize

from .backend import get_backend

import ot

def projection_simplex(V, z=1, axis=None):

r"""Projection of :math:`\mathbf{V}` onto the simplex, scaled by `z`

.. math::

P\left(\mathbf{V}, z\right) = \mathop{\arg \min}_{\substack{\mathbf{y} >= 0 \\ \sum_i \mathbf{y}_i = z}} \quad \|\mathbf{y} - \mathbf{V}\|^2

Parameters

----------

V: ndarray, rank 2

z: float or array

If array, len(z) must be compatible with :math:`\mathbf{V}`

axis: None or int

- axis=None: project :math:`\mathbf{V}` by :math:`P(\mathbf{V}.\mathrm{ravel}(), z)`

- axis=1: project each :math:`\mathbf{V}_i` by :math:`P(\mathbf{V}_i, z_i)`

- axis=0: project each :math:`\mathbf{V}_{:, j}` by :math:`P(\mathbf{V}_{:, j}, z_j)`

Returns

-------

projection: ndarray, shape :math:`\mathbf{V}`.shape

"""

if axis == 1:

n_features = V.shape[1]

U = np.sort(V, axis=1)[:, ::-1]

z = np.ones(len(V)) * z

cssv = np.cumsum(U, axis=1) - z[:, np.newaxis]

ind = np.arange(n_features) + 1

cond = U - cssv / ind > 0

rho = np.count_nonzero(cond, axis=1)

theta = cssv[np.arange(len(V)), rho - 1] / rho

return np.maximum(V - theta[:, np.newaxis], 0)

elif axis == 0:

return projection_simplex(V.T, z, axis=1).T

else:

V = V.ravel().reshape(1, -1)

return projection_simplex(V, z, axis=1).ravel()</div>

class Regularization(object):

r"""Base class for Regularization objects

Notes

-----

This class is not intended for direct use but as apparent for true

regularization implementation.

"""

def __init__(self, gamma=1.0):

"""

Parameters

----------

gamma: float

Regularization parameter.

We recover unregularized OT when gamma -> 0.

"""

self.gamma = gamma

def delta_Omega(X):

r"""

Compute :math:`\delta_\Omega(\mathbf{X}_{:, j})` for each :math:`\mathbf{X}_{:, j}`.

.. math::

\delta_\Omega(\mathbf{x}) = \sup_{\mathbf{y} >= 0} \

\mathbf{y}^T \mathbf{x} - \Omega(\mathbf{y})

Parameters

----------

X: array, shape = (len(a), len(b))

Input array.

Returns

-------

v: array, (len(b), )

Values: :math:`\mathbf{v}_j = \delta_\Omega(\mathbf{X}_{:, j})`

G: array, (len(a), len(b))

Gradients: :math:`\mathbf{G}_{:, j} = \nabla \delta_\Omega(\mathbf{X}_{:, j})`

"""

raise NotImplementedError</div>

def max_Omega(X, b):

r"""

Compute :math:`\mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})` for each :math:`\mathbf{X}_{:, j}`.

.. math::

\mathrm{max}_{\Omega, j}(\mathbf{x}) =

\sup_{\substack{\mathbf{y} >= 0 \ \sum_i \mathbf{y}_i = 1}}

\mathbf{y}^T \mathbf{x} - \frac{1}{\mathbf{b}_j} \Omega(\mathbf{b}_j \mathbf{y})

Parameters

----------

X: array, shape = (len(a), len(b))

Input array.

b: array, shape = (len(b), )

Returns

-------

v: array, (len(b), )

Values: :math:`\mathbf{v}_j = \mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})`

G: array, (len(a), len(b))

Gradients: :math:`\mathbf{G}_{:, j} = \nabla \mathrm{max}_{\Omega, j}(\mathbf{X}_{:, j})`

"""

raise NotImplementedError</div>

def Omega(T):

"""

Compute regularization term.

Parameters

----------

T: array, shape = len(a) x len(b)

Input array.

Returns

-------

value: float

Regularization term.

"""

raise NotImplementedError</div>

</div>

class NegEntropy(Regularization):

"""NegEntropy regularization"""

def delta_Omega(self, X):

G = np.exp(X / self.gamma - 1)

return val, G</div>

def max_Omega(self, X, b):

max_X = np.max(X, axis=0) / self.gamma

exp_X = np.exp(X / self.gamma - max_X)

val = self.gamma * (np.log(np.sum(exp_X, axis=0)) + max_X)

val -= self.gamma * np.log(b)

G = exp_X / np.sum(exp_X, axis=0)

return val, G</div>

def Omega(self, T):

return self.gamma * np.sum(T * np.log(T))</div>

</div>

class SquaredL2(Regularization):

"""Squared L2 regularization"""

max_X = np.maximum(X, 0)

val = np.sum(max_X**2, axis=0) / (2 * self.gamma)

G = max_X / self.gamma

return val, G</div>

G = projection_simplex(X / (b * self.gamma), axis=0)

val -= 0.5 * self.gamma * b * np.sum(G * G, axis=0)

return val, G</div>

return 0.5 * self.gamma * np.sum(T**2)</div>

</div>

class SparsityConstrained(Regularization):

"""Squared L2 regularization with sparsity constraints"""

def __init__(self, max_nz, gamma=1.0):

self.max_nz = max_nz

self.gamma = gamma

# For each column of X, find entries that are not among the top max_nz.

non_top_indices = np.argpartition(-X, self.max_nz, axis=0)[self.max_nz :]

# Set these entries to -inf.

if X.ndim == 1:

X[non_top_indices] = 0.0

else:

X[non_top_indices, np.arange(X.shape[1])] = 0.0

G = max_X / self.gamma

return val, G</div>

# Project the scaled X onto the simplex with sparsity constraint.

G = ot.utils.projection_sparse_simplex(

X / (b * self.gamma), self.max_nz, axis=0

)

return val, G</div>

</div>

def dual_obj_grad(alpha, beta, a, b, C, regul):

r"""

Compute objective value and gradients of dual objective.

Parameters

----------

alpha: array, shape = len(a)

beta: array, shape = len(b)

Current iterate of dual potentials.

a: array, shape = len(a)

b: array, shape = len(b)

Input histograms (should be non-negative and sum to 1).

C: array, shape = (len(a), len(b))

Ground cost matrix.

regul: Regularization object

Should implement a `delta_Omega(X)` method.

Returns

-------

obj: float

Objective value (higher is better).

grad_alpha: array, shape = len(a)

Gradient w.r.t. `alpha`.

grad_beta: array, shape = len(b)

Gradient w.r.t. `beta`.

"""

obj = np.dot(alpha, a) + np.dot(beta, b)

grad_alpha = a.copy()

grad_beta = b.copy()

# X[:, j] = alpha + beta[j] - C[:, j]

X = alpha[:, np.newaxis] + beta - C

# val.shape = len(b)

# G.shape = len(a) x len(b)

val, G = regul.delta_Omega(X)

obj -= np.sum(val)

grad_alpha -= G.sum(axis=1)

grad_beta -= G.sum(axis=0)

return obj, grad_alpha, grad_beta</div>

def solve_dual(

a, b, C, regul, method="L-BFGS-B", tol=1e-3, max_iter=500, verbose=False

):

"""

Solve the "smoothed" dual objective.

Parameters

----------

a: array, shape = (len(a), )

b: array, shape = (len(b), )

Input histograms (should be non-negative and sum to 1).

C: array, shape = (len(a), len(b))

Ground cost matrix.

regul: Regularization object

Should implement a `delta_Omega(X)` method.

method: str

Solver to be used (passed to `scipy.optimize.minimize`).

tol: float

Tolerance parameter.

max_iter: int

Maximum number of iterations.

Returns

-------

alpha: array, shape = (len(a), )

beta: array, shape = (len(b), )

Dual potentials.

"""

def _func(params):

# Unpack alpha and beta.

alpha = params[: len(a)]

beta = params[len(a) :]

obj, grad_alpha, grad_beta = dual_obj_grad(alpha, beta, a, b, C, regul)

# Pack grad_alpha and grad_beta.

grad = np.concatenate((grad_alpha, grad_beta))

# We need to maximize the dual.

return -obj, -grad

# Unfortunately, `minimize` only supports functions whose argument is a

# vector. So, we need to concatenate alpha and beta.

alpha_init = np.zeros(len(a))

beta_init = np.zeros(len(b))

params_init = np.concatenate((alpha_init, beta_init))

res = minimize(

_func,

params_init,

method=method,

jac=True,

tol=tol,

options=dict(maxiter=max_iter, disp=verbose),

)

alpha = res.x[: len(a)]

beta = res.x[len(a) :]

return alpha, beta, res</div>

def semi_dual_obj_grad(alpha, a, b, C, regul):

"""

Compute objective value and gradient of semi-dual objective.

Parameters

----------

alpha: array, shape = len(a)

Current iterate of semi-dual potentials.

a: array, shape = len(a)

b: array, shape = len(b)

Input histograms (should be non-negative and sum to 1).

C: array, shape = (len(a), len(b))

Ground cost matrix.

regul: Regularization object

Should implement a `max_Omega(X)` method.

Returns

-------

obj: float

Objective value (higher is better).

grad: array, shape = len(a)

Gradient w.r.t. alpha.

"""

grad = a.copy()

# X[:, j] = alpha - C[:, j]

# val.shape = len(b)

# G.shape = len(a) x len(b)

val, G = regul.max_Omega(X, b)

obj -= np.dot(b, val)

grad -= np.dot(G, b)

return obj, grad</div>

def solve_semi_dual(

):

"""

Solve the "smoothed" semi-dual objective.

Parameters

----------

a: array, shape = (len(a), )

b: array, shape = (len(b), )

Input histograms (should be non-negative and sum to 1).

C: array, shape = (len(a), len(b))

Ground cost matrix.

regul: Regularization object

Should implement a `max_Omega(X)` method.

method: str

Solver to be used (passed to `scipy.optimize.minimize`).

tol: float

Tolerance parameter.

max_iter: int

Maximum number of iterations.

Returns

-------

alpha: array, shape = (len(a), )

Semi-dual potentials.

"""

def _func(alpha):

obj, grad = semi_dual_obj_grad(alpha, a, b, C, regul)

# We need to maximize the semi-dual.

return -obj, -grad

res = minimize(

_func,

alpha_init,

method=method,

jac=True,

tol=tol,

)

return res.x, res</div>

def get_plan_from_dual(alpha, beta, C, regul):

r"""

Retrieve optimal transportation plan from optimal dual potentials.

Parameters

----------

alpha: array, shape = len(a)

beta: array, shape = len(b)

Optimal dual potentials.

C: array, shape = (len(a), len(b))

Ground cost matrix.

regul: Regularization object

Should implement a `delta_Omega(X)` method.

Returns

-------

T: array, shape = (len(a), len(b))

Optimal transportation plan.

"""

return regul.delta_Omega(X)[1]</div>

def get_plan_from_semi_dual(alpha, b, C, regul):

r"""

Retrieve optimal transportation plan from optimal semi-dual potentials.

Parameters

----------

alpha: array, shape = len(a)

Optimal semi-dual potentials.

b: array, shape = len(b)

Second input histogram (should be non-negative and sum to 1).

C: array, shape = (len(a), len(b))

Ground cost matrix.

regul: Regularization object

Should implement a `delta_Omega(X)` method.

Returns

-------

T: array, shape = (len(a), len(b))

Optimal transportation plan.

"""

return regul.max_Omega(X, b)[1] * b</div>

def smooth_ot_dual(

a,

b,

M,

reg,

reg_type="l2",

method="L-BFGS-B",

stopThr=1e-9,

numItermax=500,

verbose=False,

log=False,

max_nz=None,

):

r"""

Solve the regularized OT problem in the dual and return the OT matrix

The function solves the smooth relaxed dual formulation (7) in

:ref:`[17] <references-smooth-ot-dual>`:

.. math::

\max_{\alpha,\beta}\quad \mathbf{a}^T\alpha + \mathbf{b}^T\beta -

\sum_j \delta_\Omega \left(\alpha+\beta_j-\mathbf{m}_j \right)

where :

- :math:`\mathbf{m}_j` is the j-th column of the cost matrix

- :math:`\delta_\Omega` is the convex conjugate of the regularization term :math:`\Omega`

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

The OT matrix can is reconstructed from the gradient of :math:`\delta_\Omega`

(See :ref:`[17] <references-smooth-ot-dual>` Proposition 1).

The optimization algorithm is using gradient decent (L-BFGS by default).

Parameters

----------

a : np.ndarray (ns,)

samples weights in the source domain

b : np.ndarray (nt,) or np.ndarray (nt,nbb)

samples in the target domain, compute sinkhorn with multiple targets

and fixed :math:`\mathbf{M}` if :math:`\mathbf{b}` is a matrix

(return OT loss + dual variables in log)

M : np.ndarray (ns,nt)

loss matrix

reg : float

Regularization term >0

reg_type : str

Regularization type, can be the following (default ='l2'):

- 'kl' : Kullback Leibler (~ Neg-entropy used in sinkhorn

:ref:`[2] <references-smooth-ot-dual>`)

- 'l2' : Squared Euclidean regularization

- 'sparsity_constrained' : Sparsity-constrained regularization [50]

max_nz : int or None, optional. Used only in the case of reg_type = 'sparsity_constrained' to specify the maximum number of nonzeros per column of the optimal plan;

not used for other regularization types.

method : str

Solver to use for scipy.optimize.minimize

numItermax : int, optional

Max number of iterations

stopThr : float, optional

Stop threshold on error (>0)

verbose : bool, optional

Print information along iterations

log : bool, optional

record log if True

Returns

-------

gamma : (ns, nt) ndarray

Optimal transportation matrix for the given parameters

log : dict

log dictionary return only if log==True in parameters

.. _references-smooth-ot-dual:

References

----------

.. [2] M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport,

Advances in Neural Information Processing Systems (NIPS) 26, 2013

.. [17] Blondel, M., Seguy, V., & Rolet, A. (2018). Smooth and Sparse Optimal Transport.

Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics (AISTATS).

.. [50] Liu, T., Puigcerver, J., & Blondel, M. (2023). Sparsity-constrained optimal transport.

Proceedings of the Eleventh International Conference on Learning Representations (ICLR).

See Also

--------

ot.lp.emd : Unregularized OT

ot.sinhorn : Entropic regularized OT

ot.optim.cg : General regularized OT

"""

nx = get_backend(a, b, M)

if reg_type.lower() in ["l2", "squaredl2"]:

regul = SquaredL2(gamma=reg)

elif reg_type.lower() in ["entropic", "negentropy", "kl"]:

regul = NegEntropy(gamma=reg)

elif reg_type.lower() in ["sparsity_constrained", "sparsity-constrained"]:

if not isinstance(max_nz, int):

raise ValueError(f"max_nz {max_nz} must be an integer")

regul = SparsityConstrained(gamma=reg, max_nz=max_nz)

else:

raise NotImplementedError("Unknown regularization")

a0, b0, M0 = a, b, M

# convert to humpy

a, b, M = nx.to_numpy(a, b, M)

# solve dual

alpha, beta, res = solve_dual(

a, b, M, regul, max_iter=numItermax, tol=stopThr, verbose=verbose

)

# reconstruct transport matrix

G = nx.from_numpy(get_plan_from_dual(alpha, beta, M, regul), type_as=M0)

if log:

log = {

"alpha": nx.from_numpy(alpha, type_as=a0),

"beta": nx.from_numpy(beta, type_as=b0),

"res": res,

}

return G, log

else:

return G</div>

def smooth_ot_semi_dual(

a,

b,

M,

reg,

reg_type="l2",

max_nz=None,

method="L-BFGS-B",

stopThr=1e-9,

numItermax=500,

verbose=False,

log=False,

):

r"""

Solve the regularized OT problem in the semi-dual and return the OT matrix

The function solves the smooth relaxed dual formulation (10) in

:ref:`[17] <references-smooth-ot-semi-dual>`:

.. math::

\max_{\alpha}\quad \mathbf{a}^T\alpha- \mathrm{OT}_\Omega^*(\alpha, \mathbf{b})

where :

.. math::

\mathrm{OT}_\Omega^*(\alpha,b)=\sum_j \mathbf{b}_j

- :math:`\mathbf{m}_j` is the j-th column of the cost matrix

- :math:`\mathrm{OT}_\Omega^*(\alpha,b)` is defined in Eq. (9) in

:ref:`[17] <references-smooth-ot-semi-dual>`

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

The OT matrix can is reconstructed using :ref:`[17] <references-smooth-ot-semi-dual>` Proposition 2.

The optimization algorithm is using gradient decent (L-BFGS by default).

Parameters

----------

a : np.ndarray (ns,)

samples weights in the source domain

b : np.ndarray (nt,) or np.ndarray (nt,nbb)

samples in the target domain, compute sinkhorn with multiple targets

and fixed:math:`\mathbf{M}` if :math:`\mathbf{b}` is a matrix

(return OT loss + dual variables in log)

M : np.ndarray (ns,nt)

loss matrix

reg : float

Regularization term >0

reg_type : str

Regularization type, can be the following (default ='l2'):

- 'kl' : Kullback Leibler (~ Neg-entropy used in sinkhorn

:ref:`[2] <references-smooth-ot-semi-dual>`)

- 'l2' : Squared Euclidean regularization

- 'sparsity_constrained' : Sparsity-constrained regularization [50]

max_nz : int or None, optional. Used only in the case of reg_type = 'sparsity_constrained' to specify the maximum number of nonzeros per column of the optimal plan;

not used for other regularization types.

method : str

Solver to use for scipy.optimize.minimize

numItermax : int, optional

Max number of iterations

stopThr : float, optional

Stop threshold on error (>0)

verbose : bool, optional

Print information along iterations

log : bool, optional

record log if True

Returns

-------

gamma : (ns, nt) ndarray

Optimal transportation matrix for the given parameters

log : dict

log dictionary return only if log==True in parameters

.. _references-smooth-ot-semi-dual:

References

----------

.. [2] M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport,

Advances in Neural Information Processing Systems (NIPS) 26, 2013

.. [17] Blondel, M., Seguy, V., & Rolet, A. (2018). Smooth and Sparse Optimal Transport.

Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics (AISTATS).

.. [50] Liu, T., Puigcerver, J., & Blondel, M. (2023). Sparsity-constrained optimal transport.

Proceedings of the Eleventh International Conference on Learning Representations (ICLR).

See Also

--------

ot.lp.emd : Unregularized OT

ot.sinhorn : Entropic regularized OT

ot.optim.cg : General regularized OT

"""

elif reg_type.lower() in ["sparsity_constrained", "sparsity-constrained"]:

else:

raise NotImplementedError("Unknown regularization")

# solve dual

alpha, res = solve_semi_dual(

)

# reconstruct transport matrix

G = get_plan_from_semi_dual(alpha, b, M, regul)

if log:

log = {"alpha": alpha, "res": res}

return G, log

else:

return G</div>

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