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<!DOCTYPE html>

<head>

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

</head>

POT Python Optimal Transport

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

"""

Stochastic solvers for regularized OT.

"""

# Authors: Kilian Fatras <kilian.fatras@gmail.com>

# Rémi Flamary <remi.flamary@polytechnique.edu>

#

# License: MIT License

import numpy as np

from .utils import dist, check_random_state

from .backend import get_backend

##############################################################################

# Optimization toolbox for SEMI - DUAL problems

##############################################################################

def coordinate_grad_semi_dual(b, M, reg, beta, i):

r"""

Compute the coordinate gradient update for regularized discrete distributions for :math:`(i, :)`

The function computes the gradient of the semi dual problem:

.. math::

\max_\mathbf{v} \ \sum_i \mathbf{a}_i \left[ \sum_j \mathbf{v}_j \mathbf{b}_j - \mathrm{reg}

\cdot \log \left( \sum_j \mathbf{b}_j

\exp \left( \frac{\mathbf{v}_j - \mathbf{M}_{i,j}}{\mathrm{reg}}

\right) \right) \right]

Where :

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

- :math:`\mathbf{v}` is a dual variable in :math:`\mathbb{R}^{nt}`

- reg is the regularization term

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

The algorithm used for solving the problem is the ASGD & SAG algorithms

as proposed in :ref:`[18] <references-coordinate-grad-semi-dual>` [alg.1 & alg.2]

Parameters

----------

b : ndarray, shape (nt,)

Target measure.

M : ndarray, shape (ns, nt)

Cost matrix.

reg : float

Regularization term > 0.

v : ndarray, shape (nt,)

Dual variable.

i : int

Picked number `i`.

Returns

-------

coordinate gradient : ndarray, shape (nt,)

.. _references-coordinate-grad-semi-dual:

References

----------

.. [18] Genevay, A., Cuturi, M., Peyré, G. & Bach, F. (2016)

Stochastic Optimization for Large-scale Optimal Transport.

Advances in Neural Information Processing Systems (2016).

"""

r = M[i, :] - beta

exp_beta = np.exp(-r / reg) * b

khi = exp_beta / (np.sum(exp_beta))

return b - khi</div>

def sag_entropic_transport(a, b, M, reg, numItermax=10000, lr=None, random_state=None):

r"""

Compute the SAG algorithm to solve the regularized discrete measures optimal transport max problem

The function solves the following optimization problem:

.. math::

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

\mathrm{reg} \cdot\Omega(\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 with :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`

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

The algorithm used for solving the problem is the SAG algorithm

as proposed in :ref:`[18] <references-sag-entropic-transport>` [alg.1]

Parameters

----------

a : ndarray, shape (ns,),

Source measure.

b : ndarray, shape (nt,),

Target measure.

M : ndarray, shape (ns, nt),

Cost matrix.

reg : float

Regularization term > 0

numItermax : int

Number of iteration.

lr : float

Learning rate.

random_state : int, RandomState instance or None, default=None

Determines random number generation. Pass an int for reproducible

output across multiple function calls.

Returns

-------

v : ndarray, shape (`nt`,)

Dual variable.

.. _references-sag-entropic-transport:

References

----------

.. [18] Genevay, A., Cuturi, M., Peyré, G. & Bach, F. (2016)

Stochastic Optimization for Large-scale Optimal Transport.

Advances in Neural Information Processing Systems (2016).

"""

if lr is None:

lr = 1.0 / max(a / reg)

n_source = np.shape(M)[0]

n_target = np.shape(M)[1]

cur_beta = np.zeros(n_target)

stored_gradient = np.zeros((n_source, n_target))

sum_stored_gradient = np.zeros(n_target)

rng = check_random_state(random_state)

for _ in range(numItermax):

i = rng.randint(n_source)

cur_coord_grad = a[i] * coordinate_grad_semi_dual(b, M, reg, cur_beta, i)

sum_stored_gradient += cur_coord_grad - stored_gradient[i]

stored_gradient[i] = cur_coord_grad

cur_beta += lr * (1.0 / n_source) * sum_stored_gradient

return cur_beta</div>

def averaged_sgd_entropic_transport(

a, b, M, reg, numItermax=300000, lr=None, random_state=None

):

r"""

Compute the ASGD algorithm to solve the regularized semi continuous measures optimal transport max problem

The function solves the following optimization problem:

.. math::

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

\mathrm{reg}\cdot\Omega(\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 with :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`

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

The algorithm used for solving the problem is the ASGD algorithm

as proposed in :ref:`[18] <references-averaged-sgd-entropic-transport>` [alg.2]

Parameters

----------

a : ndarray, shape (ns,)

source measure

b : ndarray, shape (nt,)

target measure

M : ndarray, shape (ns, nt)

cost matrix

reg : float

Regularization term > 0

numItermax : int

Number of iteration.

lr : float

Learning rate.

random_state : int, RandomState instance or None, default=None

Determines random number generation. Pass an int for reproducible

output across multiple function calls.

Returns

-------

ave_v : ndarray, shape (`nt`,)

dual variable

.. _references-averaged-sgd-entropic-transport:

References

----------

.. [18] Genevay, A., Cuturi, M., Peyré, G. & Bach, F. (2016)

Stochastic Optimization for Large-scale Optimal Transport.

Advances in Neural Information Processing Systems (2016).

"""

if lr is None:

ave_beta = np.zeros(n_target)

rng = check_random_state(random_state)

for cur_iter in range(numItermax):

k = cur_iter + 1

cur_coord_grad = coordinate_grad_semi_dual(b, M, reg, cur_beta, i)

cur_beta += (lr / np.sqrt(k)) * cur_coord_grad

ave_beta = (1.0 / k) * cur_beta + (1 - 1.0 / k) * ave_beta

return ave_beta</div>

def c_transform_entropic(b, M, reg, beta):

r"""

The goal is to recover u from the c-transform.

The function computes the c-transform of a dual variable from the other

dual variable:

.. math::

\mathbf{u} = \mathbf{v}^{c,reg} = - \mathrm{reg} \sum_j \mathbf{b}_j

\exp\left( \frac{\mathbf{v} - \mathbf{M}}{\mathrm{reg}} \right)

Where :

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

- :math:`\mathbf{u}`, :math:`\mathbf{v}` are dual variables in :math:`\mathbb{R}^{ns} \times \mathbb{R}^{nt}`

- reg is the regularization term

It is used to recover an optimal u from optimal v solving the semi dual

problem, see Proposition 2.1 of :ref:`[18] <references-c-transform-entropic>`

Parameters

----------

b : ndarray, shape (nt,)

Target measure

M : ndarray, shape (ns, nt)

Cost matrix

reg : float

Regularization term > 0

v : ndarray, shape (nt,)

Dual variable.

Returns

-------

u : ndarray, shape (`ns`,)

Dual variable.

.. _references-c-transform-entropic:

References

----------

.. [18] Genevay, A., Cuturi, M., Peyré, G. & Bach, F. (2016)

Stochastic Optimization for Large-scale Optimal Transport.

Advances in Neural Information Processing Systems (2016).

"""

alpha = np.zeros(n_source)

for i in range(n_source):

min_r = np.min(r)

exp_beta = np.exp(-(r - min_r) / reg) * b

alpha[i] = min_r - reg * np.log(np.sum(exp_beta))

return alpha</div>

def solve_semi_dual_entropic(

a, b, M, reg, method, numItermax=10000, lr=None, log=False

):

r"""

Compute the transportation matrix to solve the regularized discrete measures optimal transport max problem

The function solves the following optimization problem:

.. math::

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

\mathrm{reg} \cdot\Omega(\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 with :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`

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

The algorithm used for solving the problem is the SAG or ASGD algorithms

as proposed in :ref:`[18] <references-solve-semi-dual-entropic>`

Parameters

----------

a : ndarray, shape (ns,)

source measure

b : ndarray, shape (nt,)

target measure

M : ndarray, shape (ns, nt)

cost matrix

reg : float

Regularization term > 0

method : str

used method (SAG or ASGD)

numItermax : int

number of iteration

lr : float

learning rate

n_source : int

size of the source measure

n_target : int

size of the target measure

log : bool, optional

record log if True

Returns

-------

pi : ndarray, shape (ns, nt)

transportation matrix

log : dict

log dictionary return only if log==True in parameters

.. _references-solve-semi-dual-entropic:

References

----------

.. [18] Genevay, A., Cuturi, M., Peyré, G. & Bach, F. (2016)

Stochastic Optimization for Large-scale Optimal Transport.

Advances in Neural Information Processing Systems (2016).

"""

if method.lower() == "sag":

opt_beta = sag_entropic_transport(a, b, M, reg, numItermax, lr)

elif method.lower() == "asgd":

opt_beta = averaged_sgd_entropic_transport(a, b, M, reg, numItermax, lr)

else:

print("Please, select your method between SAG and ASGD")

return None

opt_alpha = c_transform_entropic(b, M, reg, opt_beta)

pi = (

np.exp((opt_alpha[:, None] + opt_beta[None, :] - M[:, :]) / reg)

* a[:, None]

* b[None, :]

)

if log:

log = {}

log["alpha"] = opt_alpha

log["beta"] = opt_beta

return pi, log

else:

return pi</div>

##############################################################################

# Optimization toolbox for DUAL problems

##############################################################################

def batch_grad_dual(a, b, M, reg, alpha, beta, batch_size, batch_alpha, batch_beta):

r"""

Computes the partial gradient of the dual optimal transport problem.

For each :math:`(i,j)` in a batch of coordinates, the partial gradients are :

.. math::

\partial_{\mathbf{u}_i} F = \frac{b_s}{l_v} \mathbf{u}_i -

\sum_{j \in B_v} \mathbf{a}_i \mathbf{b}_j

\exp\left( \frac{\mathbf{u}_i + \mathbf{v}_j - \mathbf{M}_{i,j}}{\mathrm{reg}} \right)

\partial_{\mathbf{v}_j} F = \frac{b_s}{l_u} \mathbf{v}_j -

\sum_{i \in B_u} \mathbf{a}_i \mathbf{b}_j

\exp\left( \frac{\mathbf{u}_i + \mathbf{v}_j - \mathbf{M}_{i,j}}{\mathrm{reg}} \right)

Where :

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

- :math:`\mathbf{u}`, :math:`\mathbf{v}` are dual variables in :math:`\mathbb{R}^{ns} \times \mathbb{R}^{nt}`

- reg is the regularization term

- :math:`B_u` and :math:`B_v` are lists of index

- :math:`b_s` is the size of the batches :math:`B_u` and :math:`B_v`

- :math:`l_u` and :math:`l_v` are the lengths of :math:`B_u` and :math:`B_v`

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

The algorithm used for solving the dual problem is the SGD algorithm

as proposed in :ref:`[19] <references-batch-grad-dual>` [alg.1]

Parameters

----------

a : ndarray, shape (ns,)

source measure

b : ndarray, shape (nt,)

target measure

M : ndarray, shape (ns, nt)

cost matrix

reg : float

Regularization term > 0

alpha : ndarray, shape (ns,)

dual variable

beta : ndarray, shape (nt,)

dual variable

batch_size : int

size of the batch

batch_alpha : ndarray, shape (bs,)

batch of index of alpha

batch_beta : ndarray, shape (bs,)

batch of index of beta

Returns

-------

grad : ndarray, shape (`ns`,)

partial grad F

.. _references-batch-grad-dual:

References

----------

.. [19] Seguy, V., Bhushan Damodaran, B., Flamary, R., Courty, N., Rolet, A.& Blondel, M.

Large-scale Optimal Transport and Mapping Estimation. International Conference on Learning Representation (2018)

"""

G = -(

np.exp(

(

alpha[batch_alpha, None]

+ beta[None, batch_beta]

- M[batch_alpha, :][:, batch_beta]

)

/ reg

)

* a[batch_alpha, None]

* b[None, batch_beta]

)

grad_beta = np.zeros(np.shape(M)[1])

grad_alpha = np.zeros(np.shape(M)[0])

grad_beta[batch_beta] = b[batch_beta] * len(batch_alpha) / np.shape(M)[0] + G.sum(0)

grad_alpha[batch_alpha] = a[batch_alpha] * len(batch_beta) / np.shape(M)[1] + G.sum(

1

)

return grad_alpha, grad_beta</div>

def sgd_entropic_regularization(

a, b, M, reg, batch_size, numItermax, lr, random_state=None

):

r"""

Compute the sgd algorithm to solve the regularized discrete measures optimal transport dual problem

The function solves the following optimization problem:

.. math::

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

\mathrm{reg} \cdot\Omega(\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 with :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`

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

Parameters

----------

a : ndarray, shape (ns,)

source measure

b : ndarray, shape (nt,)

target measure

M : ndarray, shape (ns, nt)

cost matrix

reg : float

Regularization term > 0

batch_size : int

size of the batch

numItermax : int

number of iteration

lr : float

learning rate

random_state : int, RandomState instance or None, default=None

Determines random number generation. Pass an int for reproducible

output across multiple function calls.

Returns

-------

alpha : ndarray, shape (ns,)

dual variable

beta : ndarray, shape (nt,)

dual variable

References

----------

.. [19] Seguy, V., Bhushan Damodaran, B., Flamary, R., Courty, N., Rolet, A.& Blondel, M. Large-scale Optimal Transport and Mapping Estimation. International Conference on Learning Representation (2018)

"""

cur_alpha = np.zeros(n_source)

rng = check_random_state(random_state)

for cur_iter in range(numItermax):

k = np.sqrt(cur_iter + 1)

batch_alpha = rng.choice(n_source, batch_size, replace=False)

batch_beta = rng.choice(n_target, batch_size, replace=False)

update_alpha, update_beta = batch_grad_dual(

a, b, M, reg, cur_alpha, cur_beta, batch_size, batch_alpha, batch_beta

)

cur_alpha[batch_alpha] += (lr / k) * update_alpha[batch_alpha]

cur_beta[batch_beta] += (lr / k) * update_beta[batch_beta]

return cur_alpha, cur_beta</div>

def solve_dual_entropic(a, b, M, reg, batch_size, numItermax=10000, lr=1, log=False):

r"""

Compute the transportation matrix to solve the regularized discrete measures optimal transport dual problem

The function solves the following optimization problem:

.. math::

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

\mathrm{reg} \cdot\Omega(\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 with :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`

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

Parameters

----------

a : ndarray, shape (ns,)

source measure

b : ndarray, shape (nt,)

target measure

M : ndarray, shape (ns, nt)

cost matrix

reg : float

Regularization term > 0

batch_size : int

size of the batch

numItermax : int

number of iteration

lr : float

learning rate

log : bool, optional

record log if True

Returns

-------

pi : ndarray, shape (ns, nt)

transportation matrix

log : dict

log dictionary return only if log==True in parameters

References

----------

"""

opt_alpha, opt_beta = sgd_entropic_regularization(

)

pi = (

* a[:, None]

* b[None, :]

)

if log:

log = {}

log["alpha"] = opt_alpha

log["beta"] = opt_beta

return pi, log

else:

return pi</div>

################################################################################

# Losses for stochastic optimization

################################################################################

def loss_dual_entropic(u, v, xs, xt, reg=1, ws=None, wt=None, metric="sqeuclidean"):

r"""

Compute the dual loss of the entropic OT as in equation (6)-(7) of [19]

This loss is backend compatible and can be used for stochastic optimization

of the dual potentials. It can be used on the full dataset (beware of

memory) or on minibatches.

Parameters

----------

u : array-like, shape (ns,)

Source dual potential

v : array-like, shape (nt,)

Target dual potential

xs : array-like, shape (ns,d)

Source samples

xt : array-like, shape (ns,d)

Target samples

reg : float

Regularization term > 0 (default=1)

ws : array-like, shape (ns,), optional

Source sample weights (default unif)

wt : array-like, shape (ns,), optional

Target sample weights (default unif)

metric : string, callable

Ground metric for OT (default quadratic). Can be given as a callable

function taking (xs,xt) as parameters.

Returns

-------

dual_loss : array-like

Dual loss (to maximize)

References

----------

"""

nx = get_backend(u, v, xs, xt)

if ws is None:

ws = nx.ones(xs.shape[0], type_as=xs) / xs.shape[0]

if wt is None:

wt = nx.ones(xt.shape[0], type_as=xt) / xt.shape[0]

if callable(metric):

M = metric(xs, xt)

else:

M = dist(xs, xt, metric=metric)

F = -reg * nx.exp((u[:, None] + v[None, :] - M) / reg)

return nx.sum(u * ws) + nx.sum(v * wt) + nx.sum(ws[:, None] * F * wt[None, :])</div>

def plan_dual_entropic(u, v, xs, xt, reg=1, ws=None, wt=None, metric="sqeuclidean"):

r"""

Compute the primal OT plan the entropic OT as in equation (8) of [19]

This loss is backend compatible and can be used for stochastic optimization

of the dual potentials. It can be used on the full dataset (beware of

memory) or on minibatches.

Parameters

----------

u : array-like, shape (ns,)

Source dual potential

v : array-like, shape (nt,)

Target dual potential

xs : array-like, shape (ns,d)

Source samples

xt : array-like, shape (ns,d)

Target samples

reg : float

Regularization term > 0 (default=1)

ws : array-like, shape (ns,), optional

Source sample weights (default unif)

wt : array-like, shape (ns,), optional

Target sample weights (default unif)

metric : string, callable

Ground metric for OT (default quadratic). Can be given as a callable

function taking (xs,xt) as parameters.

Returns

-------

G : array-like

Primal OT plan

References

----------

"""

if ws is None:

if wt is None:

if callable(metric):

else:

H = nx.exp((u[:, None] + v[None, :] - M) / reg)

return ws[:, None] * H * wt[None, :]</div>

def loss_dual_quadratic(u, v, xs, xt, reg=1, ws=None, wt=None, metric="sqeuclidean"):

r"""

Compute the dual loss of the quadratic regularized OT as in equation (6)-(7) of [19]

This loss is backend compatible and can be used for stochastic optimization

of the dual potentials. It can be used on the full dataset (beware of

memory) or on minibatches.

Parameters

----------

u : array-like, shape (ns,)

Source dual potential

v : array-like, shape (nt,)

Target dual potential

xs : array-like, shape (ns,d)

Source samples

xt : array-like, shape (ns,d)

Target samples

reg : float

Regularization term > 0 (default=1)

ws : array-like, shape (ns,), optional

Source sample weights (default unif)

wt : array-like, shape (ns,), optional

Target sample weights (default unif)

metric : string, callable

Ground metric for OT (default quadratic). Can be given as a callable

function taking (xs,xt) as parameters.

Returns

-------

dual_loss : array-like

Dual loss (to maximize)

References

----------

"""

if ws is None:

if wt is None:

if callable(metric):

else:

F = -1.0 / (4 * reg) * nx.maximum(u[:, None] + v[None, :] - M, 0.0) ** 2

def plan_dual_quadratic(u, v, xs, xt, reg=1, ws=None, wt=None, metric="sqeuclidean"):

r"""

Compute the primal OT plan the quadratic regularized OT as in equation (8) of [19]

This loss is backend compatible and can be used for stochastic optimization

of the dual potentials. It can be used on the full dataset (beware of

memory) or on minibatches.

Parameters

----------

u : array-like, shape (ns,)

Source dual potential

v : array-like, shape (nt,)

Target dual potential

xs : array-like, shape (ns,d)

Source samples

xt : array-like, shape (ns,d)

Target samples

reg : float

Regularization term > 0 (default=1)

ws : array-like, shape (ns,), optional

Source sample weights (default unif)

wt : array-like, shape (ns,), optional

Target sample weights (default unif)

metric : string, callable

Ground metric for OT (default quadratic). Can be given as a callable

function taking (xs,xt) as parameters.

Returns

-------

G : array-like

Primal OT plan

References

----------

"""

if ws is None:

if wt is None:

if callable(metric):

else:

H = 1.0 / (2 * reg) * nx.maximum(u[:, None] + v[None, :] - M, 0.0)

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