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<!DOCTYPE html>

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

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

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

"""

Utility functions for batch operations in optimal transport.

"""

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

# Paul Krzakala <paul.krzakala@gmail.com>

# Thibaut Germain <thibaut.germain.pro@gmail.com>

#

# License: MIT License

from ot.backend import get_backend

def entropy_batch(T, nx=None, eps=1e-16):

r"""Computes the entropy of a batch of transport plans T.

.. math::

H(T)_b = - \sum_{i,j} T_{b,i,j} \log(T_{b,i,j})

Parameters

----------

T : array-like, shape (b,n1,n2)

`b` transport of size `n1` x `n2`.

eps : float, optional

Small constant to avoid numerical issues with log(0). Default is 1e-16.

nx : Backend, optional

Backend to perform computations on. If omitted, the backend defaults to that of `T`.

"""

if nx is None:

nx = get_backend(T)

return -nx.sum(T * nx.log(T + eps), axis=(1, 2))</div>

def norm_batch(u, p=2, nx=None):

"""Computes the lp norm of a batch of vectors u."""

if nx is None:

nx = get_backend(u)

return nx.sum(u**p, axis=1) ** (1 / p)

def bmv(A, b, nx):

"""

Batched matrix-vector multiplication for tensor A and vector b.

"""

return nx.einsum("bij,bj->bi", A, b)

def bop(a, b, nx):

"""

Batched outer product for vectors a and b.

"""

return nx.einsum("bi,bj->bij", a, b)

def bregman_projection_batch(

K, a=None, b=None, nx=None, max_iter=10000, tol=1e-5, grad="detach"

):

r"""

Apply Bregman projection to a batch of affinity matrices :math:`\mathbf{K}`.

The function solves the following optimization problem:

.. math::

\begin{aligned}

\mathbf{T} = \mathop{\arg \min}_\mathbf{T} \quad & \text{KL}(\mathbf{T} \| \mathbf{K}) \\

\text{s.t.} \quad & \mathbf{T} \mathbf{1} = \mathbf{a} \\

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

& \mathbf{T} \geq 0

\end{aligned}

This is equivalent to:

.. math::

\begin{aligned}

\mathbf{T} = \mathop{\arg \max}_\mathbf{T} \quad & \langle \mathbf{T}, \log(\mathbf{K}) \rangle_F \\

\text{s.t.} \quad & \mathbf{T} \mathbf{1} = \mathbf{a} \\

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

& \mathbf{T} \geq 0

\end{aligned}

The optimal solution has the form :math:`\mathbf{T} = \text{diag}(\mathbf{f}) \mathbf{K} \text{diag}(\mathbf{g})`,

where the dual variables :math:`\mathbf{u}` and :math:`\mathbf{v}` are found iteratively using:

.. math::

\mathbf{f}^{(k+1)} = \frac{\mathbf{a}}{\sum \mathbf{K} \mathbf{g}^{(k)}}

\mathbf{g}^{(k+1)} = \frac{\mathbf{b}}{\sum \mathbf{K}^T \mathbf{f}^{(k)}}

Parameters

----------

K : array-like, shape (B, n, m)

Affinity matrix for each problem in the batch.

a : array-like, shape (B, n), optional

Source distribution for each problem. If None, uniform distribution is used.

b : array-like, shape (B, m), optional

Target distribution for each problem. If None, uniform distribution is used.

nx : backend object, optional

Numerical backend to use for computations. If None, the default backend is used.

max_iter : int, optional

Maximum number of iterations.

tol : float, optional

Tolerance for convergence. The solver stops when the maximum change in

the dual variables is below this value.

grad : str, optional

Gradient computation mode: 'detach', 'autodiff', or 'last_step'.

Returns

-------

dict

Dictionary containing:

- 'T' : array-like, shape (B, n, m)

Optimal transport plan for each problem.

- 'potentials' : tuple of array-like, shapes ((B, n), (B, m))

Scaling factors (f, g).

- 'n_iters' : int

Number of iterations performed.

Examples

--------

>>> import numpy as np

>>> from ot.batch import bregman_projection_batch

>>> # Create batch of affinity matrices

>>> K = np.random.randn(5, 10, 15) # 5 problems, 10x15 cost matrices

>>> result = bregman_projection_batch(K)

>>> T = result['T'] # Shape (5, 10, 15)

See Also

--------

ot.batch.bregman_log_projection_batch : Bregman projection in the log-domain.

"""

if nx is None:

nx = get_backend(a, b, K)

B, n, m = K.shape

if a is None:

a = nx.ones((B, n)) / n

if b is None:

b = nx.ones((B, m)) / m

if grad == "detach":

K = nx.detach(K)

elif grad == "last_step":

K_, K = K.clone(), nx.detach(K)

f = nx.ones((B, n)) # a / nx.sum(K, axis=2)

g = nx.ones((B, m)) # b / nx.sum(K, axis=1)

for n_iters in range(max_iter):

f = a / nx.sum(K * g[:, None, :], axis=2)

g = b / nx.sum(K * f[:, :, None], axis=1)

if n_iters % 10 == 0:

T = K * f[:, :, None] * g[:, None, :]

marginal = nx.sum(T, axis=2)

err = nx.max(norm_batch(marginal - a))

if err < tol:

break

if grad == "last_step":

summand_f = nx.sum(K_ * g[:, None, :], axis=2)

f = a / summand_f

summand_g = nx.sum(K_ * f[:, :, None], axis=1)

g = b / summand_g

potentials = (f, g)

out = {"T": T, "n_iters": n_iters, "potentials": potentials}

return out</div>

def bregman_log_projection_batch(

):

r"""

Apply Bregman projection to a batch of affinity matrices :math:`\exp(\mathbf{K})`.

The function solves the following optimization problem:

.. math::

\begin{aligned}

\mathbf{T} = \mathop{\arg \min}_\mathbf{T} \quad & \text{KL}(\mathbf{T} \| \exp(\mathbf{K})) \\

\text{s.t.} \quad & \mathbf{T} \mathbf{1} = \mathbf{a} \\

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

& \mathbf{T} \geq 0

\end{aligned}

This is equivalent to:

.. math::

\begin{aligned}

\mathbf{T} = \mathop{\arg \max}_\mathbf{T} \quad & \langle \mathbf{T}, \mathbf{K} \rangle_F \\

\text{s.t.} \quad & \mathbf{T} \mathbf{1} = \mathbf{a} \\

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

& \mathbf{T} \geq 0

\end{aligned}

The optimal solution has the form :math:`\mathbf{T} = \text{diag}(\exp(\mathbf{u})) \exp(\mathbf{K}) \text{diag}(\exp(\mathbf{v}))`,

where the dual variables :math:`\mathbf{u}` and :math:`\mathbf{v}` are found iteratively using:

.. math::

\mathbf{u}^{(k+1)} = \log(\mathbf{a}) - \text{LSE}(\mathbf{K} + \mathbf{v}^{(k)})

\mathbf{v}^{(k+1)} = \log(\mathbf{b}) - \text{LSE}(\mathbf{K}^T + \mathbf{u}^{(k)})

where LSE denotes the log-sum-exp operation. The iterations are performed using

the log-sum-exp trick to avoid numerical issues.

Parameters

----------

K : array-like, shape (B, n, m)

Affinity matrix for each problem in the batch.

a : array-like, shape (B, n), optional

Source distribution for each problem. If None, uniform distribution is used.

b : array-like, shape (B, m), optional

Target distribution for each problem. If None, uniform distribution is used.

nx : backend object, optional

Numerical backend to use for computations. If None, the default backend is used.

max_iter : int, optional

Maximum number of iterations.

tol : float, optional

Tolerance for convergence. The solver stops when the maximum change in

the dual variables is below this value.

grad : str, optional

Gradient computation mode: 'detach', 'autodiff', or 'last_step'.

Returns

-------

dict

Dictionary containing:

- 'T' : array-like, shape (B, n, m)

Optimal transport plan for each problem.

- 'log_T' : array-like, shape (B, n, m)

Logarithm of the optimal transport plan for each problem.

- 'potentials' : tuple of array-like, shapes ((B, n), (B, m))

Log-scaling factors (u, v).

- 'n_iters' : int

Number of iterations performed.

Examples

--------

>>> import numpy as np

>>> from ot.batch import bregman_log_projection_batch

>>> # Create batch of affinity matrices

>>> K = np.random.randn(5, 10, 15) # 5 problems, 10x15 cost matrices

>>> result = bregman_log_projection_batch(K)

>>> T = result['T'] # Shape (5, 10, 15)

See Also

--------

ot.batch.bregman_projection_batch : standard Bregman projection.

"""

if nx is None:

if a is None:

if b is None:

u = nx.zeros((B, n), type_as=K) # u = nx.log(a) - nx.logsumexp(K, axis=2).squeeze()

v = nx.zeros((B, m), type_as=K) # v = nx.log(b) - nx.logsumexp(K, axis=1).squeeze()

loga = nx.log(a)

logb = nx.log(b)

if grad == "detach":

elif grad == "last_step":

for n_iters in range(max_iter):

u = loga - nx.logsumexp(K + v[:, None, :], axis=2)

v = logb - nx.logsumexp(K + u[:, :, None], axis=1)

# Check convergence once every 10 iterations

if n_iters % 10 == 0:

T = nx.exp(K + u[:, :, None] + v[:, None, :])

if err < tol:

break

if grad == "last_step":

summand_u = K_ + v[:, None, :]

u = loga - nx.logsumexp(summand_u, axis=2)

summand_v = K_ + u[:, :, None]

v = logb - nx.logsumexp(summand_v, axis=1)

log_T = K + u[:, :, None] + v[:, None, :]

log_potentials = (u, v)

return {"T": T, "log_T": log_T, "n_iters": n_iters, "potentials": log_potentials}</div>

def proximal_bregman_log_plan_batch(

C,

a=None,

b=None,

nx=None,

reg=None,

inner_reg=1e-3,

max_iter=10000,

tol=1e-5,

inner_iter=1,

grad="detach",

):

r"""

Returns optimal transport plans for a batch of cost matrices :math:`\mathbf{C}` using a Bregman divergence based proximal point method :ref:`[92] <references-OT-bregman-proximal-point>`.

This function solves the following optimization problem:

.. math::

\begin{aligned}

\mathbf{T} = \mathop{\arg \min}_\mathbf{T} \quad & \langle \mathbf{T}, \mathbf{C} \rangle_F + \textit{reg} \cdot \sum_{i,j} \mathbf{T}_{i,j} \log \mathbf{T}_{i,j} \\

\text{s.t.} \quad & \mathbf{T} \mathbf{1} = \mathbf{a} \\

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

& \mathbf{T} \geq 0

\end{aligned}

The optimal transport plans are computed iteratively with a proximal point method based on a Bregman divergence where each iteration involves solving a Bregman projection problem:

.. math::

\mathbf{T}^{(k+1)} = \mathop{\arg \min}_\mathbf{T} \quad \langle \mathbf{C} - \textit{inner\_reg} \cdot \log \mathbf{T}^{(k)}, \mathbf{T} \rangle + (\textit{reg} + \textit{inner\_reg}) \cdot \sum_{i,j} \mathbf{T}_{i,j} \log \mathbf{T}_{i,j}

Denoting :math:`\mathbf{K}^{(k)} = - (\mathbf{C} + \textit{inner\_reg} \cdot \log \mathbf{T}^{(k)})/(\textit{reg} + \textit{inner\_reg})`, the affinity matrix at iteration :math:`k`, the Bregman projection problem is solved in the log-domain with a finite number of inner iterations :math:`\text{inner\_iter}`, i.e., the dual variables :math:`\mathbf{u}` and :math:`\mathbf{v}` are updated as follows:

.. math::

\mathbf{u}^{(i+1)} = \log(\mathbf{a}) - \text{LSE}(\mathbf{K}^{(k)} + \mathbf{v}^{(i)})

\mathbf{v}^{(i+1)} = \log(\mathbf{b}) - \text{LSE}((\mathbf{K}^{(k)})^T + \mathbf{u}^{(i)})

where LSE denotes the log-sum-exp operation.

Parameters

----------

C : array-like, shape (B, n, m)

Cost matrix for each problem in the batch.

a : array-like, shape (B, n), optional

Source distribution for each problem. If None, uniform distribution is used.

b : array-like, shape (B, m), optional

Target distribution for each problem. If None, uniform distribution is used.

nx : backend object, optional

Numerical backend to use for computations. If None, the default backend is used.

max_iter : int, optional

Maximum number of iterations.

inner_iter : int, optional

Number of inner iterations for updating the dual variables u and v. Default is 1.

inner_reg : float, optional

Regularization parameter for the Bregman divergence. Default is 1e-3.

tol : float, optional

Tolerance for convergence. The solver stops when the maximum change in

the dual variables is below this value.

grad : str, optional

Gradient computation mode: 'detach', 'autodiff', or 'last_step'.

Returns

-------

dict

Dictionary containing:

- 'T' : array-like, shape (B, n, m)

Optimal transport plan for each problem.

- 'log_T' : array-like, shape (B, n, m)

Logarithm of the optimal transport plan for each problem.

- 'potentials' : tuple of array-like, shapes ((B, n), (B, m))

Log-scaling factors (u, v).

- 'n_iters' : int

Number of iterations performed.

Example

--------

>>> import numpy as np

>>> from ot.batch import proximal_bregman_log_plan_batch

>>> # Create batch of affinity matrices

>>> C = np.random.rand(5, 10, 15) # 5 problems, 10x15 cost matrices

>>> result = proximal_bregman_log_plan_batch(C)

>>> T = result['T'] # Shape (5, 10, 15)

.. _references-OT-bregman-proximal-point:

Reference

----------

.. [92] Xie, Y., Wang, X., Wang, R., & Zha, H. (2020, August).

A fast proximal point method for computing exact wasserstein distance.

In Uncertainty in artificial intelligence (pp. 433-453). PMLR.

"""

if nx is None:

B, n, m = C.shape

if a is None:

if b is None:

if reg is None:

reg = 0.0

if grad == "detach":

C = nx.detach(C)

elif grad == "last_step":

C_, C = C.clone(), nx.detach(C)

log_T = nx.zeros(C.shape, type_as=C)

for n_iters in range(max_iter):

K_proj = -(C + inner_reg * log_T) / (reg + inner_reg)

for _ in range(inner_iter):

u = loga - nx.logsumexp(K_proj + v[:, None, :], axis=2)

v = logb - nx.logsumexp(K_proj + u[:, :, None], axis=1)

log_T = K_proj + u[:, :, None] + v[:, None, :]

# Check convergence once every 10 iterations

if n_iters % 10 == 0:

if err < tol:

break

if grad == "last_step":

K_proj = -(C_ + inner_reg * log_T) / (reg + inner_reg)

for _ in range(inner_iter):

u = loga - nx.logsumexp(K_proj + v[:, None, :], axis=2)

v = logb - nx.logsumexp(K_proj + u[:, :, None], axis=1)

log_potentials = (u, v)

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