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<!DOCTYPE html>

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

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

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

"""

Solvers for the original linear program OT problem.

"""

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

#

# License: MIT License

import numpy as np

import warnings

from ..utils import list_to_array, check_number_threads

from ..backend import get_backend

from .emd_wrap import emd_c, emd_c_sparse, emd_c_lazy, check_result

def center_ot_dual(alpha0, beta0, a=None, b=None):

r"""Center dual OT potentials w.r.t. their weights

The main idea of this function is to find unique dual potentials

that ensure some kind of centering/fairness. The main idea is to find dual potentials that lead to the same final objective value for both source and targets (see below for more details). It will help having

stability when multiple calling of the OT solver with small changes.

Basically we add another constraint to the potential that will not

change the objective value but will ensure unicity. The constraint

is the following:

.. math::

\alpha^T \mathbf{a} = \beta^T \mathbf{b}

in addition to the OT problem constraints.

since :math:`\sum_i a_i=\sum_j b_j` this can be solved by adding/removing

a constant from both :math:`\alpha_0` and :math:`\beta_0`.

.. math::

c &= \frac{\beta_0^T \mathbf{b} - \alpha_0^T \mathbf{a}}{\mathbf{1}^T \mathbf{b} + \mathbf{1}^T \mathbf{a}}

\alpha &= \alpha_0 + c

\beta &= \beta_0 + c

Parameters

----------

alpha0 : (ns, ...) numpy.ndarray, float64

Source dual potential

beta0 : (nt, ...) numpy.ndarray, float64

Target dual potential

a : (ns, ...) numpy.ndarray, float64

Source histogram (uniform weight if empty list)

b : (nt, ...) numpy.ndarray, float64

Target histogram (uniform weight if empty list)

Returns

-------

alpha : (ns, ...) numpy.ndarray, float64

Source centered dual potential

beta : (nt, ...) numpy.ndarray, float64

Target centered dual potential

"""

nx = get_backend(alpha0, beta0, a, b)

n = alpha0.shape[0]

m = beta0.shape[0]

# if no weights are provided, use uniform

if a is None:

a = nx.full(alpha0.shape, 1.0 / n, type_as=alpha0)

elif a.ndim != alpha0.ndim:

a = nx.repeat(a[..., None], alpha0.shape[-1], -1)

if b is None:

b = nx.full(beta0.shape, 1.0 / m, type_as=beta0)

elif b.ndim != beta0.ndim:

b = nx.repeat(b[..., None], beta0.shape[-1], -1)

# compute constant that balances the weighted sums of the duals

ips = nx.sum(b * beta0, axis=0) - nx.sum(a * alpha0, axis=0)

denom = nx.sum(a, axis=0) + nx.sum(b, axis=0)

c = ips / denom

# update duals

alpha = alpha0 + c

beta = beta0 - c

return alpha, beta

def estimate_dual_null_weights(alpha0, beta0, a, b, M):

r"""Estimate feasible values for 0-weighted dual potentials

The feasible values are computed efficiently but rather coarsely.

.. warning::

This function is necessary because the C++ solver in `emd_c`

discards all samples in the distributions with

zeros weights. This means that while the primal variable (transport

matrix) is exact, the solver only returns feasible dual potentials

on the samples with weights different from zero.

First we compute the constraints violations:

.. math::

\mathbf{V} = \alpha + \beta^T - \mathbf{M}

Next we compute the max amount of violation per row (:math:`\alpha`) and

columns (:math:`beta`)

.. math::

\mathbf{v^a}_i = \max_j \mathbf{V}_{i,j}

\mathbf{v^b}_j = \max_i \mathbf{V}_{i,j}

Finally we update the dual potential with 0 weights if a

constraint is violated

.. math::

\alpha_i = \alpha_i - \mathbf{v^a}_i \quad \text{ if } \mathbf{a}_i=0 \text{ and } \mathbf{v^a}_i>0

\beta_j = \beta_j - \mathbf{v^b}_j \quad \text{ if } \mathbf{b}_j=0 \text{ and } \mathbf{v^b}_j > 0

In the end the dual potentials are centered using function

:py:func:`ot.lp.center_ot_dual`.

Note that all those updates do not change the objective value of the

solution but provide dual potentials that do not violate the constraints.

Parameters

----------

alpha0 : (ns,) numpy.ndarray, float64

Source dual potential

beta0 : (nt,) numpy.ndarray, float64

Target dual potential

alpha0 : (ns,) numpy.ndarray, float64

Source dual potential

beta0 : (nt,) numpy.ndarray, float64

Target dual potential

a : (ns,) numpy.ndarray, float64

Source distribution (uniform weights if empty list)

b : (nt,) numpy.ndarray, float64

Target distribution (uniform weights if empty list)

M : (ns,nt) numpy.ndarray, float64

Loss matrix (c-order array with type float64)

Returns

-------

alpha : (ns,) numpy.ndarray, float64

Source corrected dual potential

beta : (nt,) numpy.ndarray, float64

Target corrected dual potential

"""

# binary indexing of non-zeros weights

asel = a != 0

bsel = b != 0

# compute dual constraints violation

constraint_violation = alpha0[:, None] + beta0[None, :] - M

# Compute largest violation per line and columns

aviol = np.max(constraint_violation, 1)

bviol = np.max(constraint_violation, 0)

# update corrects violation of

alpha_up = -1 * ~asel * np.maximum(aviol, 0)

beta_up = -1 * ~bsel * np.maximum(bviol, 0)

alpha = alpha0 + alpha_up

beta = beta0 + beta_up

return center_ot_dual(alpha, beta, a, b)

def _compute_active_subset(a, b, M, row_mask, col_mask):

"""Return filtered inputs restricted to the rows/columns with mass."""

need_filter = np.any(~row_mask) or np.any(~col_mask)

if not need_filter:

return need_filter, None, None, a, b, M

row_idx = np.flatnonzero(row_mask)

col_idx = np.flatnonzero(col_mask)

M_solver = np.asarray(M[np.ix_(row_idx, col_idx)], dtype=np.float64, order="C")

return need_filter, row_idx, col_idx, a[row_idx], b[col_idx], M_solver

def _inflate_from_active_subset(

G, u, v, need_filter, row_idx, col_idx, row_mask, col_mask, n_rows, n_cols

):

"""Embed the filtered dense solution back into the full support."""

if not need_filter:

return G, u, v

G_full = np.zeros((n_rows, n_cols), dtype=G.dtype)

G_full[np.ix_(row_idx, col_idx)] = G

u_full = np.zeros((n_rows,), dtype=u.dtype)

v_full = np.zeros((n_cols,), dtype=v.dtype)

u_full[row_mask] = u

v_full[col_mask] = v

return G_full, u_full, v_full

def _prepare_warmstart(potentials_init, need_filter, row_mask, col_mask):

"""Return warm-start potentials filtered to the active support."""

if potentials_init is None:

return None, None

alpha_init, beta_init = potentials_init

alpha_init = np.asarray(alpha_init, dtype=np.float64)

beta_init = np.asarray(beta_init, dtype=np.float64)

if need_filter:

alpha_init = alpha_init[row_mask]

beta_init = beta_init[col_mask]

return alpha_init, beta_init

def emd(

a,

b,

M,

numItermax=100000,

log=False,

center_dual=True,

numThreads=1,

check_marginals=True,

potentials_init=None,

):

r"""Solves the Earth Movers distance problem and returns the OT matrix

.. math::

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

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

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

\gamma \geq 0

where :

- :math:`\mathbf{M}` is the metric cost matrix

- :math:`\mathbf{a}` and :math:`\mathbf{b}` are the sample weights

.. warning:: Note that the :math:`\mathbf{M}` matrix in numpy needs to be a C-order

numpy.array in float64 format. It will be converted if not in this

format

.. note:: This function is backend-compatible and will work on arrays

from all compatible backends. But the algorithm uses the C++ CPU backend

which can lead to copy overhead on GPU arrays.

.. note:: This function will cast the computed transport plan to the data type

of the provided input with the following priority: :math:`\mathbf{a}`,

then :math:`\mathbf{b}`, then :math:`\mathbf{M}` if marginals are not provided.

Casting to an integer tensor might result in a loss of precision.

If this behaviour is unwanted, please make sure to provide a

floating point input.

.. note:: An error will be raised if the vectors :math:`\mathbf{a}` and :math:`\mathbf{b}` do not sum to the same value.

Uses the algorithm proposed in :ref:`[1] <references-emd>`.

Parameters

----------

a : (ns,) array-like, float

Source histogram (uniform weight if empty list)

b : (nt,) array-like, float

Target histogram (uniform weight if empty list)

M : (ns,nt) array-like or sparse matrix, float

Loss matrix. Can be:

- Dense array (c-order array in numpy with type float64)

- Sparse matrix in backend's format (scipy.sparse.coo_matrix for NumPy backend,

torch.sparse_coo_tensor for PyTorch backend, etc.)

numItermax : int, optional (default=100000)

The maximum number of iterations before stopping the optimization

algorithm if it has not converged.

log: bool, optional (default=False)

If True, returns a dictionary containing the cost and dual variables.

Otherwise returns only the optimal transportation matrix.

center_dual: boolean, optional (default=True)

If True, centers the dual potential using function

:py:func:`ot.lp.center_ot_dual`.

numThreads: int or "max", optional (default=1)

Deprecated compatibility parameter. The network simplex solver no

longer uses OpenMP, so this parameter is ignored.

check_marginals: bool, optional (default=True)

If True, checks that the marginals mass are equal. If False, skips the

check.

potentials_init: tuple of two arrays (alpha, beta), optional (default=None)

Warmstart dual potentials to accelerate convergence. Should be a tuple

(alpha, beta) where alpha is shape (ns,) and beta is shape (nt,).

These potentials are used to guide initial pivots in the network simplex.

Typically obtained from a previous EMD solve or Sinkhorn approximation.

.. note:: The solver automatically detects sparse format using the backend's

:py:meth:`issparse` method. For sparse inputs:

- Uses a memory-efficient sparse EMD algorithm

- Returns the transport plan as a sparse matrix in the backend's format

- Supports scipy.sparse (NumPy), torch.sparse (PyTorch), etc.

- JAX and TensorFlow backends don't support sparse matrices

Returns

-------

gamma: array-like or sparse matrix, shape (ns, nt)

Optimal transportation matrix for the given parameters.

- For dense inputs: returns a dense array

- For sparse inputs: returns a sparse matrix in the backend's format

(e.g., scipy.sparse.coo_matrix for NumPy, torch.sparse_coo_tensor for PyTorch)

log: dict, optional

If input log is True, a dictionary containing the cost, dual variables,

and exit status.

Examples

--------

Simple example with obvious solution. The function emd accepts lists and

perform automatic conversion to numpy arrays

>>> import ot

>>> a=[.5,.5]

>>> b=[.5,.5]

>>> M=[[0.,1.],[1.,0.]]

>>> ot.emd(a, b, M)

array([[0.5, 0. ],

[0. , 0.5]])

.. _references-emd:

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.

See Also

--------

ot.bregman.sinkhorn : Entropic regularized OT

ot.optim.cg : General regularized OT

"""

n1, n2 = None, None

# Convert lists to arrays, using M to detect backend when a,b are empty

a, b, M = list_to_array(a, b, M)

nx = get_backend(a, b, M)

# Check if M is sparse using backend's issparse method

is_sparse = nx.issparse(M)

if is_sparse:

# Check if backend supports sparse matrices

backend_name = nx.__class__.__name__

if backend_name in ["JaxBackend", "TensorflowBackend"]:

raise NotImplementedError(

f"Sparse optimal transport is not supported for {backend_name}. "

"JAX does not have native sparse matrix support, and TensorFlow's "

"sparse implementation is incomplete. Please convert your sparse "

"matrix to dense format using M.toarray() or equivalent before calling emd()."

)

# Extract COO data using backend method - returns numpy arrays

edge_sources, edge_targets, edge_costs, (n1, n2) = nx.sparse_coo_data(M)

# Ensure correct dtypes for C++ solver

if edge_sources.dtype != np.uint64:

edge_sources = edge_sources.astype(np.uint64)

if edge_targets.dtype != np.uint64:

edge_targets = edge_targets.astype(np.uint64)

if edge_costs.dtype != np.float64:

edge_costs = edge_costs.astype(np.float64)

if a is not None and len(a) != 0:

type_as = a

elif b is not None and len(b) != 0:

type_as = b

else:

type_as = a if a is not None else b

# Set n1, n2 if not already set (dense case)

if n1 is None:

n1, n2 = M.shape

if a is None or len(a) == 0:

a = nx.ones((n1,), type_as=type_as) / n1

if b is None or len(b) == 0:

b = nx.ones((n2,), type_as=type_as) / n2

if is_sparse:

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

else:

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

M = np.asarray(M, dtype=np.float64, order="C")

a = np.asarray(a, dtype=np.float64)

b = np.asarray(b, dtype=np.float64)

assert (

a.shape[0] == n1 and b.shape[0] == n2

), "Dimension mismatch, check dimensions of M with a and b"

# ensure that same mass

if check_marginals:

np.testing.assert_almost_equal(

a.sum(0),

b.sum(0),

err_msg="a and b vector must have the same sum",

decimal=6,

)

asel = a != 0

bsel = b != 0

numThreads = check_number_threads(numThreads)

if numThreads != 1:

warnings.warn(

"The 'numThreads' parameter is deprecated and will be removed in a "

"future version. The network simplex solver no longer uses OpenMP, "

"so this parameter is ignored.",

DeprecationWarning,

stacklevel=2,

)

# ============================================================================

# CALL SOLVER (sparse or dense)

# ============================================================================

if is_sparse:

# Sparse solver - never build full matrix

flow_sources, flow_targets, flow_values, cost, u, v, result_code = emd_c_sparse(

a, b, edge_sources, edge_targets, edge_costs, numItermax

)

else:

row_mask = asel

col_mask = bsel

(

need_filter,

row_idx,

col_idx,

a_solver,

b_solver,

M_solver,

) = _compute_active_subset(a, b, M, row_mask, col_mask)

alpha_init, beta_init = _prepare_warmstart(

potentials_init, need_filter, row_mask, col_mask

)

# Dense solver

G, cost, u, v, result_code = emd_c(

a_solver, b_solver, M_solver, numItermax, numThreads, alpha_init, beta_init

)

G, u, v = _inflate_from_active_subset(

G, u, v, need_filter, row_idx, col_idx, row_mask, col_mask, n1, n2

)

# ============================================================================

# POST-PROCESS DUAL VARIABLES AND CREATE TRANSPORT PLAN

# ============================================================================

# Center dual potentials

if center_dual:

u, v = center_ot_dual(u, v, a, b)

# Handle null weights

if np.any(~asel) or np.any(~bsel):

if is_sparse:

else:

u, v = estimate_dual_null_weights(u, v, a, b, M)

result_code_string = check_result(result_code)

# Create transport plan in backend format

if is_sparse:

# Convert flow to sparse matrix using backend's coo_matrix method

flow_values_backend = nx.from_numpy(flow_values, type_as=type_as)

flow_sources_backend = nx.from_numpy(

flow_sources.astype(np.int64), type_as=type_as

)

flow_targets_backend = nx.from_numpy(

flow_targets.astype(np.int64), type_as=type_as

)

G = nx.coo_matrix(

flow_values_backend,

flow_sources_backend,

flow_targets_backend,

shape=(n1, n2),

type_as=type_as,

)

else:

# Warn about integer casting for dense case

if not nx.is_floating_point(type_as):

warnings.warn(

"Input histogram consists of integer. The transport plan will be "

"casted accordingly, possibly resulting in a loss of precision. "

"If this behaviour is unwanted, please make sure your input "

"histogram consists of floating point elements.",

stacklevel=2,

)

G = nx.from_numpy(G, type_as=type_as)

# Return results

if log:

log_dict = {

"cost": cost,

"u": nx.from_numpy(u, type_as=type_as),

"v": nx.from_numpy(v, type_as=type_as),

"warning": result_code_string,

"result_code": result_code,

}

return G, log_dict

else:

return G</div>

def emd2(

a,

b,

M,

processes=1,

numItermax=100000,

log=False,

return_matrix=False,

center_dual=True,

numThreads=1,

check_marginals=True,

potentials_init=None,

):

r"""Solves the Earth Movers distance problem and returns the loss

.. math::

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

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

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

\gamma \geq 0

where :

- :math:`\mathbf{M}` is the metric cost matrix

- :math:`\mathbf{a}` and :math:`\mathbf{b}` are the sample weights

.. note:: This function is backend-compatible and will work on arrays

from all compatible backends. But the algorithm uses the C++ CPU backend

which can lead to copy overhead on GPU arrays.

.. note:: This function will cast the computed transport plan and

transportation loss to the data type of the provided input with the

following priority : :math:`\mathbf{a}`, then :math:`\mathbf{b}`,

then :math:`\mathbf{M}` if marginals are not provided.

Casting to an integer tensor might result in a loss of precision.

If this behaviour is unwanted, please make sure to provide a

floating point input.

.. note:: An error will be raised if the vectors :math:`\mathbf{a}` and :math:`\mathbf{b}` do not sum to the same value.

Uses the algorithm proposed in :ref:`[1] <references-emd2>`.

Parameters

----------

a : (ns,) array-like, float64

Source histogram (uniform weight if empty list)

b : (nt,) array-like, float64

Target histogram (uniform weight if empty list)

M : (ns,nt) array-like or sparse matrix, float64

Loss matrix. Can be:

- Dense array (c-order array in numpy with type float64)

- Sparse matrix in backend's format (scipy.sparse.coo_matrix for NumPy backend,

torch.sparse_coo_tensor for PyTorch backend, etc.)

processes : int, optional (default=1)

Nb of processes used for multiple emd computation (deprecated)

numItermax : int, optional (default=100000)

The maximum number of iterations before stopping the optimization

algorithm if it has not converged.

log: boolean, optional (default=False)

If True, returns a dictionary containing the cost and dual

variables. Otherwise returns only the optimal transportation cost.

return_matrix: boolean, optional (default=False)

If True, returns the optimal transportation matrix in the log.

center_dual: boolean, optional (default=True)

If True, centers the dual potential using function

:py:func:`ot.lp.center_ot_dual`.

numThreads: int or "max", optional (default=1)

Deprecated compatibility parameter. The network simplex solver no

longer uses OpenMP, so this parameter is ignored.

check_marginals: bool, optional (default=True)

If True, checks that the marginals mass are equal. If False, skips the

check.

potentials_init: tuple of two arrays (alpha, beta), optional (default=None)

Warmstart dual potentials to accelerate convergence. Should be a tuple

(alpha, beta) where alpha is shape (ns,) and beta is shape (nt,).

These potentials are used to guide initial pivots in the network simplex.

Typically obtained from a previous EMD solve or Sinkhorn approximation.

.. note:: The solver automatically detects sparse format using the backend's

:py:meth:`issparse` method. For sparse inputs:

- Uses a memory-efficient sparse EMD algorithm

- Edges not included are treated as having infinite cost (forbidden)

- Supports scipy.sparse (NumPy), torch.sparse (PyTorch), etc.

- JAX and TensorFlow backends don't support sparse matrices

Returns

-------

W: float, array-like

Optimal transportation loss for the given parameters

log: dict

If input log is true, a dictionary containing the cost, dual

variables (u, v), exit status, and optionally the optimal

transportation matrix (G) if return_matrix is True

Examples

--------

Simple example with obvious solution. The function emd accepts lists and

perform automatic conversion to numpy arrays

>>> import ot

>>> a=[.5,.5]

>>> b=[.5,.5]

>>> M=[[0.,1.],[1.,0.]]

>>> ot.emd2(a,b,M)

0.0

.. _references-emd2:

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.

See Also

--------

ot.bregman.sinkhorn : Entropic regularized OT

ot.optim.cg : General regularized OT

"""

n1, n2 = None, None

# Check if M is sparse using backend's issparse method

# Save original sparse tensor for gradient tracking (before conversion to numpy)

M_original_sparse = None

if is_sparse:

# Check if backend supports sparse matrices

backend_name = nx.__class__.__name__

raise NotImplementedError()

# Save original M for gradient tracking (before numpy conversion)

M_original_sparse = M

type_as = a

type_as = b

else:

# Set n1, n2 if not already set (dense case)

if n1 is None:

n1, n2 = M.shape

# if empty array given then use uniform distributions

a0, b0 = a, b

M0 = None if is_sparse else M

if is_sparse:

# Use the original sparse tensor (preserves gradients for PyTorch)

edge_costs_original = M_original_sparse

else:

edge_costs_original = None

if is_sparse:

else:

assert (

), "Dimension mismatch, check dimensions of M with a and b"

# ensure that same mass

if check_marginals:

np.testing.assert_almost_equal(

a.sum(0),

b.sum(0, keepdims=True),

err_msg="a and b vector must have the same sum",

decimal=6,

)

b = b * a.sum(0) / b.sum(0, keepdims=True)

asel = a != 0

numThreads = check_number_threads(numThreads)

if numThreads != 1:

warnings.warn(

"The 'numThreads' parameter is deprecated and will be removed in a "

"future version. The network simplex solver no longer uses OpenMP, "

"so this parameter is ignored.",

DeprecationWarning,

stacklevel=2,

)

# ============================================================================

# DEFINE SOLVER FUNCTION

# ============================================================================

def f(b):

bsel = b != 0

if is_sparse:

# Solve sparse EMD

emd_c_sparse(a, b, edge_sources, edge_targets, edge_costs, numItermax)

)

else:

(

need_filter,

row_idx,

col_idx,

a_solver,

b_solver,

M_solver,

) = _compute_active_subset(a, b, M, asel, bsel)

alpha_init, beta_init = _prepare_warmstart(

potentials_init, need_filter, asel, bsel

)

# Solve dense EMD

a_solver,

b_solver,

M_solver,

numItermax,

numThreads,

alpha_init,

beta_init,

)

G, u, v, need_filter, row_idx, col_idx, asel, bsel, n1, n2

)

# Center dual potentials

if center_dual:

# Handle null weights

if is_sparse:

else:

# Prepare cost with gradients

if is_sparse:

# Build gradient mapping for sparse case

edge_to_idx = {

(edge_sources[k], edge_targets[k]): k for k in range(len(edge_sources))

}

grad_edge_costs = np.zeros(len(edge_costs), dtype=np.float64)

for idx in range(len(flow_sources)):

src, tgt, flow = flow_sources[idx], flow_targets[idx], flow_values[idx]

edge_idx = edge_to_idx.get((src, tgt), -1)

if edge_idx >= 0:

grad_edge_costs[edge_idx] = flow

# Convert gradient to sparse format matching edge_costs_original

grad_edge_costs_backend = nx.from_numpy(grad_edge_costs, type_as=type_as)

if nx.issparse(edge_costs_original):

# Reconstruct sparse gradient tensor with same structure as original

grad_M_sparse = nx.coo_matrix(

grad_edge_costs_backend,

nx.from_numpy(edge_sources.astype(np.int64), type_as=type_as),

nx.from_numpy(edge_targets.astype(np.int64), type_as=type_as),

shape=(n1, n2),

type_as=type_as,

)

else:

grad_M_sparse = grad_edge_costs_backend

cost = nx.set_gradients(

nx.from_numpy(cost, type_as=type_as),

(a0, b0, edge_costs_original),

(

nx.from_numpy(u - np.mean(u), type_as=type_as),

nx.from_numpy(v - np.mean(v), type_as=type_as),

grad_M_sparse,

),

)

# Build transport plan in backend sparse format

flow_sources_backend = nx.from_numpy(

)

flow_targets_backend = nx.from_numpy(

)

G_backend = nx.coo_matrix(

flow_values_backend,

flow_sources_backend,

flow_targets_backend,

shape=(n1, n2),

type_as=type_as,

)

else:

# Dense case: warn about integer casting

warnings.warn(

"Input histogram consists of integer. The transport plan will be "

"casted accordingly, possibly resulting in a loss of precision. "

"If this behaviour is unwanted, please make sure your input "

"histogram consists of floating point elements.",

stacklevel=2,

)

G_backend = nx.from_numpy(G, type_as=type_as)

cost = nx.set_gradients(

(a0, b0, M0),

(

G_backend,

),

)

check_result(result_code)

# Return results

if log or return_matrix:

log_dict = {

"cost": cost,

"warning": check_result(result_code),

"result_code": result_code,

}

if return_matrix:

log_dict["G"] = G_backend

return [cost, log_dict]

else:

return cost

if len(b.shape) == 1:

return f(b)

nb = b.shape[1]

if processes > 1:

warnings.warn(

"The 'processes' parameter has been deprecated. "

"Multiprocessing should be done outside of POT."

)

res = list(map(f, [b[:, i].copy() for i in range(nb)]))

return res</div>

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