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<!DOCTYPE html>

<head>

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

</head>

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

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

"""

General OT solvers with unified API

"""

# Author: Remi Flamary <remi.flamary@polytechnique.edu>

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

#

# License: MIT License

from ..utils import OTResult, dist, split_sample_ratio, apply_scaler

from ..lp import emd2, emd2_lazy, wasserstein_1d

from ..backend import get_backend

from ..unbalanced import mm_unbalanced, sinkhorn_knopp_unbalanced, lbfgsb_unbalanced

from ..bregman import (

sinkhorn_log,

empirical_sinkhorn2,

empirical_sinkhorn_nystroem2,

old_geomloss,

)

from ..sliced import sliced_wasserstein_distance, max_sliced_wasserstein_distance

from ..bsp import compute_bspot_bijection

from ..smooth import smooth_ot_dual

from ..gaussian import (

empirical_bures_wasserstein_distance,

empirical_bures_wasserstein_distance_hd,

)

from ..factored import factored_optimal_transport

from ..lowrank import lowrank_sinkhorn

from ..optim import cg

from warnings import warn

lst_valid_methods_solve = [

"sinkhorn", # sinkhorn for unbalanced

"sinkhorn_log", # sinkhorn for unbalanced

"mm", # unbalanced

"lbfgsb", # unbalanced

]

lst_method_solve_sample = [

"1d",

"gaussian",

"gaussian_hd",

"lowrank",

"nystroem",

"factored",

"geomloss",

"geomloss_auto",

"geomloss_tensorized",

"geomloss_online",

"geomloss_multiscale",

"sliced",

"max_sliced",

"bsp",

]

all_valid_methods_solve_sample = lst_valid_methods_solve + lst_method_solve_sample

def solve(

M,

a=None,

b=None,

reg=None,

c=None,

reg_type="KL",

unbalanced=None,

unbalanced_type="KL",

method=None,

n_threads=1,

max_iter=None,

plan_init=None,

potentials_init=None,

tol=None,

verbose=False,

grad="autodiff",

):

r"""Solve the discrete optimal transport problem and return :any:`OTResult` object

The function solves the following general optimal transport problem

.. math::

\min_{\mathbf{T}\geq 0} \quad \sum_{i,j} T_{i,j}M_{i,j} + \lambda_r R(\mathbf{T}) +

\lambda_1 U(\mathbf{T}\mathbf{1},\mathbf{a}) +

\lambda_2 U(\mathbf{T}^T\mathbf{1},\mathbf{b})

The regularization is selected with `reg` (:math:`\lambda_r`) and `reg_type`. By

default ``reg=None`` and there is no regularization. The unbalanced marginal

penalization can be selected with `unbalanced` (:math:`(\lambda_1, \lambda_2)`) and

`unbalanced_type`. By default ``unbalanced=None`` and the function

solves the exact optimal transport problem (respecting the marginals).

Parameters

----------

M : array-like, shape (dim_a, dim_b)

Loss matrix

a : array-like, shape (dim_a,), optional

Samples weights in the source domain (default is uniform)

b : array-like, shape (dim_b,), optional

Samples weights in the source domain (default is uniform)

reg : float, optional

Regularization weight :math:`\lambda_r`, by default None (no reg., exact

OT)

c : array-like, shape (dim_a, dim_b), optional (default=None)

Reference measure for the regularization.

If None, then use :math:`\mathbf{c} = \mathbf{a} \mathbf{b}^T`.

If :math:`\texttt{reg_type}=`'entropy', then :math:`\mathbf{c} = 1_{dim_a} 1_{dim_b}^T`.

reg_type : str, optional

Type of regularization :math:`R` either "KL", "L2", "entropy",

by default "KL". a tuple of functions can be provided for general

solver (see :any:`cg`). This is only used when ``reg!=None``.

unbalanced : float or indexable object of length 1 or 2

Marginal relaxation term.

If it is a scalar or an indexable object of length 1,

then the same relaxation is applied to both marginal relaxations.

The balanced OT can be recovered using :math:`unbalanced=float("inf")`.

For semi-relaxed case, use either

:math:`unbalanced=(float("inf"), scalar)` or

:math:`unbalanced=(scalar, float("inf"))`.

If unbalanced is an array,

it must have the same backend as input arrays `(a, b, M)`.

unbalanced_type : str, optional

Type of unbalanced penalization function :math:`U` either "KL", "L2",

"TV", by default "KL".

method : str, optional

Method for solving the problem when multiple algorithms are available,

default None for automatic selection.

n_threads : int, optional

Number of OMP threads for exact OT solver, by default 1

max_iter : int, optional

Maximum number of iterations, by default None (default values in each solvers)

plan_init : array-like, shape (dim_a, dim_b), optional

Initialization of the OT plan for iterative methods, by default None

potentials_init : (array-like(dim_a,),array-like(dim_b,)), optional

Initialization of the OT dual potentials for iterative methods (Sinkhorn,

unbalanced) or warmstart for exact EMD solver. For EMD, these potentials

are used to guide initial pivots in the network simplex. By default None

tol : _type_, optional

Tolerance for solution precision, by default None (default values in each solvers)

verbose : bool, optional

Print information in the solver, by default False

grad : str, optional

Type of gradient computation, either or 'autodiff', 'envelope' or 'last_step' used only for

Sinkhorn solver. By default 'autodiff' provides gradients wrt all

outputs (`plan, value, value_linear`) but with important memory cost.

'envelope' provides gradients only for `value` and and other outputs are

detached. This is useful for memory saving when only the value is needed. 'last_step' provides

gradients only for the last iteration of the Sinkhorn solver, but provides gradient for both the OT plan and the objective values.

'detach' does not compute the gradients for the Sinkhorn solver.

Returns

-------

res : OTResult()

Result of the optimization problem. The information can be obtained as follows:

- res.plan : OT plan :math:`\mathbf{T}`

- res.potentials : OT dual potentials

- res.value : Optimal value of the optimization problem

- res.value_linear : Linear OT loss with the optimal OT plan

See :any:`OTResult` for more information.

Notes

-----

The following methods are available for solving the OT problems:

- **Classical exact OT problem [1]** (default parameters) :

.. math::

\min_\mathbf{T} \quad \langle \mathbf{T}, \mathbf{M} \rangle_F

s.t. \ \mathbf{T} \mathbf{1} = \mathbf{a}

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

\mathbf{T} \geq 0

can be solved with the following code:

.. code-block:: python

res = ot.solve(M, a, b)

- **Entropic regularized OT [2]** (when ``reg!=None``):

.. math::

\min_\mathbf{T} \quad \langle \mathbf{T}, \mathbf{M} \rangle_F + \lambda R(\mathbf{T})

s.t. \ \mathbf{T} \mathbf{1} = \mathbf{a}

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

\mathbf{T} \geq 0

can be solved with the following code:

.. code-block:: python

# default is ``"KL"`` regularization (``reg_type="KL"``)

res = ot.solve(M, a, b, reg=1.0)

# or for original Sinkhorn paper formulation [2]

res = ot.solve(M, a, b, reg=1.0, reg_type='entropy')

# Use envelope theorem differentiation for memory saving

res = ot.solve(M, a, b, reg=1.0, grad='envelope') # M, a, b are torch tensors

res.value.backward() # only the value is differentiable

Note that by default the Sinkhorn solver uses automatic differentiation to

compute the gradients of the values and plan. This can be changed with the

`grad` parameter. The `envelope` mode computes the gradients only

for the value and the other outputs are detached. This is useful for

memory saving when only the gradient of value is needed.

- **Quadratic regularized OT [17]** (when ``reg!=None`` and ``reg_type="L2"``):

.. math::

\min_\mathbf{T} \quad \langle \mathbf{T}, \mathbf{M} \rangle_F + \lambda R(\mathbf{T})

s.t. \ \mathbf{T} \mathbf{1} = \mathbf{a}

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

\mathbf{T} \geq 0

can be solved with the following code:

.. code-block:: python

res = ot.solve(M,a,b,reg=1.0,reg_type='L2')

- **Unbalanced OT [41]** (when ``unbalanced!=None``):

.. math::

\min_{\mathbf{T}\geq 0} \quad \sum_{i,j} T_{i,j}M_{i,j} +

\lambda_1 U(\mathbf{T}\mathbf{1},\mathbf{a}) +

\lambda_2 U(\mathbf{T}^T\mathbf{1},\mathbf{b})

can be solved with the following code:

.. code-block:: python

# default is ``"KL"``

res = ot.solve(M,a,b,unbalanced=1.0)

# quadratic unbalanced OT

res = ot.solve(M,a,b,unbalanced=1.0,unbalanced_type='L2')

# TV = partial OT

res = ot.solve(M,a,b,unbalanced=1.0,unbalanced_type='TV')

- **Regularized unbalanced regularized OT [34]** (when ``unbalanced!=None`` and ``reg!=None``):

.. math::

\min_{\mathbf{T}\geq 0} \quad \sum_{i,j} T_{i,j}M_{i,j} + \lambda_r R(\mathbf{T}) +

\lambda_1 U(\mathbf{T}\mathbf{1},\mathbf{a}) +

\lambda_2 U(\mathbf{T}^T\mathbf{1},\mathbf{b})

can be solved with the following code:

.. code-block:: python

# default is ``"KL"`` for both

res = ot.solve(M,a,b,reg=1.0,unbalanced=1.0)

# quadratic unbalanced OT with KL regularization

res = ot.solve(M,a,b,reg=1.0,unbalanced=1.0,unbalanced_type='L2')

# both quadratic

res = ot.solve(M,a,b,reg=1.0, reg_type='L2',unbalanced=1.0,unbalanced_type='L2')

.. _references-solve:

References

----------

.. [1] Bonneel, N., Van De Panne, M., Paris, S., & Heidrich, W.

(2011, December). Displacement interpolation using Lagrangian mass

transport. In ACM Transactions on Graphics (TOG) (Vol. 30, No. 6, p.

158). ACM.

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

of Optimal Transport, Advances in Neural Information Processing

Systems (NIPS) 26, 2013

.. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016).

Scaling algorithms for unbalanced transport problems.

arXiv preprint arXiv:1607.05816.

.. [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).

.. [34] Feydy, J., Séjourné, T., Vialard, F. X., Amari, S. I., Trouvé,

A., & Peyré, G. (2019, April). Interpolating between optimal transport

and MMD using Sinkhorn divergences. In The 22nd International Conference

on Artificial Intelligence and Statistics (pp. 2681-2690). PMLR.

.. [41] Chapel, L., Flamary, R., Wu, H., Févotte, C., and Gasso, G. (2021).

Unbalanced optimal transport through non-negative penalized

linear regression. NeurIPS.

"""

# detect backend

nx = get_backend(M, a, b, c)

# create uniform weights if not given

if a is None:

a = nx.ones(M.shape[0], type_as=M) / M.shape[0]

if b is None:

b = nx.ones(M.shape[1], type_as=M) / M.shape[1]

if c is None:

c = a[:, None] * b[None, :]

if reg is None:

reg = 0

if method is not None and method not in lst_valid_methods_solve:

raise ValueError(

"Unknown method={} for solve, must be in {}".format(

method, lst_valid_methods_solve

)

# default values for solutions

potentials = None

value = None

value_linear = None

plan = None

status = None

if reg == 0: # exact OT

if unbalanced is None: # Exact balanced OT

# default values for EMD solver

if max_iter is None:

max_iter = 1000000

value_linear, log = emd2(

a,

b,

M,

numItermax=max_iter,

log=True,

return_matrix=True,

numThreads=n_threads,

potentials_init=potentials_init,

)

value = value_linear

potentials = (log["u"], log["v"])

plan = log["G"]

status = log["warning"] if log["warning"] is not None else "Converged"

elif unbalanced_type.lower() in ["kl", "l2"]: # unbalanced exact OT

# default values for exact unbalanced OT

if max_iter is None:

max_iter = 1000

if tol is None:

tol = 1e-12

plan, log = mm_unbalanced(

a,

b,

M,

reg_m=unbalanced,

c=c,

reg=reg,

div=unbalanced_type,

numItermax=max_iter,

stopThr=tol,

log=True,

verbose=verbose,

G0=plan_init,

)

value_linear = log["cost"]

value = log["total_cost"]

if max_iter is None:

max_iter = 1000

if tol is None:

tol = 1e-12

if isinstance(reg_type, str):

reg_type = reg_type.lower()

plan, log = lbfgsb_unbalanced(

a,

b,

M,

reg=reg,

reg_m=unbalanced,

c=c,

reg_div=reg_type,

regm_div=unbalanced_type,

numItermax=max_iter,

stopThr=tol,

verbose=verbose,

log=True,

G0=plan_init,

)

value_linear = log["cost"]

value = log["total_cost"]

else:

raise (

NotImplementedError(

'Unknown unbalanced_type="{}"'.format(unbalanced_type)

)

else: # regularized OT

if unbalanced is None: # Balanced regularized OT

f, df = reg_type

if max_iter is None:

max_iter = 1000

if tol is None:

tol = 1e-9

plan, log = cg(

a,

b,

M,

reg=reg,

f=f,

df=df,

numItermax=max_iter,

stopThr=tol,

log=True,

verbose=verbose,

G0=plan_init,

)

value_linear = nx.sum(M * plan)

value = log["loss"][-1]

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

if grad in [

"envelope",

"last_step",

"detach",

]: # if envelope, last_step or detach then detach the input

M0, a0, b0 = M, a, b

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

# default values for sinkhorn

if max_iter is None:

max_iter = 1000

if tol is None:

tol = 1e-9

if grad == "last_step":

if max_iter == 0:

raise ValueError(

"The maximum number of iterations must be greater than 0 when using grad=last_step."

)

max_iter = max_iter - 1

plan, log = sinkhorn_log(

a,

b,

M,

reg=reg,

numItermax=max_iter,

stopThr=tol,

log=True,

verbose=verbose,

)

potentials = (log["log_u"], log["log_v"])

# if last_step, compute the last step of the Sinkhorn algorithm with the non-detached inputs

if grad == "last_step":

loga = nx.log(a0)

logb = nx.log(b0)

v = logb - nx.logsumexp(-M0 / reg + potentials[0][:, None], 0)

u = loga - nx.logsumexp(-M0 / reg + potentials[1][None, :], 1)

plan = nx.exp(-M0 / reg + u[:, None] + v[None, :])

potentials = (u, v)

log["niter"] = max_iter + 1

log["log_u"] = u

log["log_v"] = v

log["u"] = nx.exp(u)

log["v"] = nx.exp(v)

if reg_type.lower() == "entropy":

value = value_linear + reg * nx.sum(plan * nx.log(plan + 1e-16))

else:

# stabilize kl of 0 mass

if nx.any(a == 0) or nx.any(b == 0):

a = a + 1e-15 * nx.min(a)

b = b + 1e-15 * nx.min(b)

print(

"Warning: a or b has 0 mass, adding small mass to avoid NaN in KL divergence."

)

plan, a[:, None] * b[None, :] + 1e-15

)

if grad == "envelope": # set the gradient at convergence

value = nx.set_gradients(

value,

(M0, a0, b0),

(

plan,

reg * (potentials[0] - potentials[0].mean()),

reg * (potentials[1] - potentials[1].mean()),

),

)

elif reg_type.lower() == "l2":

if max_iter is None:

max_iter = 1000

if tol is None:

tol = 1e-9

plan, log = smooth_ot_dual(

a,

b,

M,

reg=reg,

numItermax=max_iter,

stopThr=tol,

log=True,

verbose=verbose,

)

potentials = (log["alpha"], log["beta"])

else:

raise (

NotImplementedError(

'Not implemented reg_type="{}"'.format(reg_type)

)

else: # unbalanced AND regularized OT

if (

not isinstance(reg_type, tuple)

and reg_type.lower() in ["kl"]

and unbalanced_type.lower() == "kl"

):

if max_iter is None:

max_iter = 1000

if tol is None:

tol = 1e-9

plan, log = sinkhorn_knopp_unbalanced(

a,

b,

M,

reg=reg,

reg_m=unbalanced,

method=method,

reg_type=reg_type,

c=c,

warmstart=potentials_init,

numItermax=max_iter,

stopThr=tol,

verbose=verbose,

log=True,

)

value_linear = log["cost"]

value = log["total_cost"]

potentials = (log["logu"], log["logv"])

elif (

isinstance(reg_type, tuple)

or reg_type.lower() in ["kl", "l2", "entropy"]

) and unbalanced_type.lower() in ["kl", "l2", "tv"]:

if isinstance(reg_type, tuple) and nx.__name__ == "tf":

raise NotImplementedError(

"Custom regularization is not implemented for TensorFlow backend."

)

if max_iter is None:

max_iter = 1000

if tol is None:

tol = 1e-12

if isinstance(reg_type, str):

reg_type = reg_type.lower()

plan, log = lbfgsb_unbalanced(

a,

b,

M,

reg=reg,

reg_m=unbalanced,

c=c,

reg_div=reg_type,

regm_div=unbalanced_type,

numItermax=max_iter,

stopThr=tol,

verbose=verbose,

log=True,

G0=plan_init,

)

value_linear = log["cost"]

value = log["total_cost"]

else:

raise (

NotImplementedError(

'Not implemented reg_type="{}" and unbalanced_type="{}"'.format(

reg_type, unbalanced_type

)

res = OTResult(

potentials=potentials,

value=value,

value_linear=value_linear,

plan=plan,

status=status,

backend=nx,

)

return res</div>

def solve_sample(

X_a,

X_b,

a=None,

b=None,

metric="sqeuclidean",

reg=None,

c=None,

reg_type="KL",

unbalanced=None,

unbalanced_type="KL",

lazy=False,

batch_size=None,

method=None,

n_threads=1,

max_iter=None,

plan_init=None,

rank=100,

scaling=0.95,

potentials_init=None,

X_init=None,

tol=None,

verbose=False,

grad="autodiff",

random_state=None,

debias=False,

n_projections=50,

projections=None,

scaler=None,

):

r"""Solve the discrete optimal transport problem using the samples in the source and target domains.

The function solves the following general optimal transport problem

.. math::

\min_{\mathbf{T}\geq 0} \quad \sum_{i,j} T_{i,j}M_{i,j} + \lambda_r R(\mathbf{T}) +

\lambda_u U(\mathbf{T}\mathbf{1},\mathbf{a}) +

\lambda_u U(\mathbf{T}^T\mathbf{1},\mathbf{b})

where the cost matrix :math:`\mathbf{M}` is computed from the samples in the

source and target domains such that :math:`M_{i,j} = d(x_i,y_j)` where

:math:`d` is a metric (by default the squared Euclidean distance).

The regularization is selected with `reg` (:math:`\lambda_r`) and `reg_type`. By

default ``reg=None`` and there is no regularization. The unbalanced marginal

penalization can be selected with `unbalanced` (:math:`\lambda_u`) and

`unbalanced_type`. By default ``unbalanced=None`` and the function

solves the exact optimal transport problem (respecting the marginals).

Parameters

----------

X_a : array-like, shape (n_samples_a, dim)

samples in the source domain

X_b : array-like, shape (n_samples_b, dim)

samples in the target domain

a : array-like, shape (dim_a,), optional

Samples weights in the source domain (default is uniform)

b : array-like, shape (dim_b,), optional

Samples weights in the source domain (default is uniform)

reg : float, optional

Regularization weight :math:`\lambda_r`, by default None (no reg., exact

OT)

c : array-like, shape (dim_a, dim_b), optional (default=None)

Reference measure for the regularization.

If None, then use :math:`\mathbf{c} = \mathbf{a} \mathbf{b}^T`.

If `reg_type='entropy'`, then :math:`\mathbf{c} = 1_{dim_a} 1_{dim_b}^T`.

reg_type : str, optional

Type of regularization :math:`R` either "KL", "L2", "entropy", by default "KL"

unbalanced : float or indexable object of length 1 or 2

Marginal relaxation term.

If it is a scalar or an indexable object of length 1,

then the same relaxation is applied to both marginal relaxations.

The balanced OT can be recovered using `unbalanced=float("inf")`.

For semi-relaxed case, use either

`unbalanced=(float("inf"), scalar)` or

`unbalanced=(scalar, float("inf"))`.

If unbalanced is an array,

it must have the same backend as input arrays `(a, b, M)`.

unbalanced_type : str, optional

Type of unbalanced penalization function :math:`U` either "KL", "L2", "TV", by default "KL"

lazy : bool, optional

Return :any:`OTResultlazy` object to reduce memory cost when True, by

default False

batch_size : int, optional

Batch size for lazy solver, by default None (default values in each

solvers)

method : str, optional

Method for solving the problem, this can be used to select the solver

for unbalanced problems (see :any:`ot.solve`), or to select a specific

large scale solver. Available methods are:

- "1d" for 1D OT solver done in parallel for each dimension.

- "gaussian" for Gaussian OT solver that estimates mean and

covariance and solve the closed form solution)

- "gaussian_hd" for high-dimensional Gaussian OT solver with rank given by `rank` parameter .

- "lowrank" for low-rank Sinkhorn solver (see :any:`ot.lowrank`)

- "nystroem" for Nystroem Sinkhorn solver

- "factored" for factored OT solver

- "geomloss" for GeomLoss Sinkhorn solver

- "sliced" for sliced wasserstein distance (see :any:`ot.sliced`)

- "max_sliced" for max sliced wasserstein distance (see

:any:`ot.sliced`)

- "bsp" for BSP-OT solver (only for distribution with same number of

samples and uniform weights)

n_threads : int, optional

Number of OMP threads for exact OT solver, by default 1

max_iter : int, optional

Maximum number of iteration, by default None (default values in each solvers)

plan_init : array-like, shape (dim_a, dim_b), optional

Initialization of the OT plan for iterative methods, by default None

rank : int, optional

Rank of the OT matrix for lazy solvers (method='factored') or (method='nystroem'), by default 100

scaling : float, optional

Scaling factor for the epsilon scaling lazy solvers (method='geomloss'), by default 0.95

potentials_init : (array-like(dim_a,),array-like(dim_b,)), optional

Initialization of the OT dual potentials for iterative methods (Sinkhorn,

unbalanced) or warmstart for exact EMD solver. For EMD, these potentials

are used to guide initial pivots in the network simplex. By default None

tol : _type_, optional

Tolerance for solution precision, by default None (default values in each solvers)

verbose : bool, optional

Print information in the solver, by default False

grad : str, optional

Type of gradient computation, either or 'autodiff' or 'envelope' used only for

Sinkhorn solver. By default 'autodiff' provides gradients wrt all

outputs (`plan, value, value_linear`) but with important memory cost.

'envelope' provides gradients only for `value` and and other outputs are

detached. This is useful for memory saving when only the value is needed.

random_state : int, optional

The random state for sampling the components in each distribution for method='nystroem'.

debias : bool, optional

Whether to use the debiased version of the OT problem, by default False

if True, the value returned is the Sinkhorn divergence [23] but the plan is

still the Sinkhorn plan. The results for all pairs of problems and

provided in the log dictionary. if debiased='split', then X_a and X_b

are split into two halves and the debiased value is computed using the

two halves as proposed in minibatch OT [91].

n_projections : int, optional

Number of projections for sliced OT, by default 50

projections : array-like, shape (n_projections, dim), optional

Projections for sliced OT, by default None (random projections are used)

scaler : callable, optional

Function to scale the input data, following :any:`ot.utils.apply_scaler`.

by default None (no scaling).

Returns

-------

res : OTResult()

Result of the optimization problem. The information can be obtained as follows:

- res.plan : OT plan :math:`\mathbf{T}`

- res.potentials : OT dual potentials

- res.value : Optimal value of the optimization problem

- res.value_linear : Linear OT loss with the optimal OT plan

- res.lazy_plan : Lazy OT plan (when ``lazy=True`` or lazy method)

See :any:`OTResult` for more information.

Notes

-----

The following methods are available for solving the OT problems:

- **Classical exact OT problem [1]** (default parameters) :

.. math::

\min_\mathbf{T} \quad \langle \mathbf{T}, \mathbf{M} \rangle_F

s.t. \ \mathbf{T} \mathbf{1} = \mathbf{a}

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

\mathbf{T} \geq 0, M_{i,j} = d(x_i,y_j)

can be solved with the following code:

.. code-block:: python

res = ot.solve_sample(xa, xb, a, b)

# for uniform weights

res = ot.solve_sample(xa, xb)

- **Entropic regularized OT [2]** (when ``reg!=None``):

.. math::

\min_\mathbf{T} \quad \langle \mathbf{T}, \mathbf{M} \rangle_F + \lambda R(\mathbf{T})

s.t. \ \mathbf{T} \mathbf{1} = \mathbf{a}

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

\mathbf{T} \geq 0, M_{i,j} = d(x_i,y_j)

can be solved with the following code:

.. code-block:: python

# default is ``"KL"`` regularization (``reg_type="KL"``)

res = ot.solve_sample(xa, xb, a, b, reg=1.0)

# or for original Sinkhorn paper formulation [2]

res = ot.solve_sample(xa, xb, a, b, reg=1.0, reg_type='entropy')

# lazy solver of memory complexity O(n)

res = ot.solve_sample(xa, xb, a, b, reg=1.0, lazy=True, batch_size=100)

# lazy OT plan

lazy_plan = res.lazy_plan

# Use envelope theorem differentiation for memory saving

res = ot.solve_sample(xa, xb, a, b, reg=1.0, grad='envelope')

res.value.backward() # only the value is differentiable

Note that by default the Sinkhorn solver uses automatic differentiation to

compute the gradients of the values and plan. This can be changed with the

`grad` parameter. The `envelope` mode computes the gradients only

for the value and the other outputs are detached. This is useful for

memory saving when only the gradient of value is needed.

We also have a very efficient solver with compiled CPU/CUDA code using

geomloss/PyKeOps that can be used with the following code:

.. code-block:: python

# automatic solver

res = ot.solve_sample(xa, xb, a, b, reg=1.0, method='geomloss')

.. warning::

The geomloss solver is a thin wrapper around the `geomloss.ot.solve_sample`

function. The API is still under development and some features might be missing. Please refer to the `geomloss` documentation for more information.

- **Quadratic regularized OT [17]** (when ``reg!=None`` and ``reg_type="L2"``):

.. math::

\min_\mathbf{T} \quad \langle \mathbf{T}, \mathbf{M} \rangle_F + \lambda R(\mathbf{T})

s.t. \ \mathbf{T} \mathbf{1} = \mathbf{a}

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

\mathbf{T} \geq 0, M_{i,j} = d(x_i,y_j)

can be solved with the following code:

.. code-block:: python

res = ot.solve_sample(xa, xb, a, b, reg=1.0, reg_type='L2')

- **Debiased OT and Sinkhorn divergence** (when ``debias=True``):

One can compute the Sinkhorn divergence between two distributions using the

following code:

.. code-block:: python

div = ot.solve_sample(xa, xb, a, b, reg=1.0, debias=True).value

Debiasing for all existing solvers is implemented and in particular one can

recover the minibatch exact OT debiasing [91] using the following code:

.. code-block:: python

div = ot.solve_sample(xa, xb, a, b, debias='split').value

- **Unbalanced OT [41]** (when ``unbalanced!=None``):

.. math::

\min_{\mathbf{T}\geq 0} \quad \sum_{i,j} T_{i,j}M_{i,j} + \lambda_u U(\mathbf{T}\mathbf{1},\mathbf{a}) + \lambda_u U(\mathbf{T}^T\mathbf{1},\mathbf{b})

\text{with} \ M_{i,j} = d(x_i,y_j)

can be solved with the following code:

.. code-block:: python

# default is ``"KL"``

res = ot.solve_sample(xa, xb, a, b, unbalanced=1.0)

# quadratic unbalanced OT

res = ot.solve_sample(xa, xb, a, b, unbalanced=1.0,unbalanced_type='L2')

# TV = partial OT

res = ot.solve_sample(xa, xb, a, b, unbalanced=1.0,unbalanced_type='TV')

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