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<!DOCTYPE html>

<head>

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

</head>

POT Python Optimal Transport

</a>

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

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

"""

Dimension reduction with OT

.. warning::

Note that by default the module is not imported in :mod:`ot`. In order to

use it you need to explicitly import :mod:`ot.dr`

"""

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

# Minhui Huang <mhhuang@ucdavis.edu>

# Jakub Zadrozny <jakub.r.zadrozny@gmail.com>

# Antoine Collas <antoine.collas@inria.fr>

#

# License: MIT License

from scipy import linalg

try:

import autograd.numpy as np

from sklearn.decomposition import PCA

import pymanopt

import pymanopt.manifolds

import pymanopt.optimizers

except ImportError:

raise ImportError(

"Missing dependency for ot.dr. Requires autograd, pymanopt, scikit-learn. You can install with install with 'pip install POT[dr]', or 'conda install autograd pymanopt scikit-learn'"

)

from .bregman import sinkhorn as sinkhorn_bregman

from .utils import dist as dist_utils, check_random_state

def dist(x1, x2):

r"""Compute squared euclidean distance between samples (autograd)"""

x1p2 = np.sum(np.square(x1), 1)

x2p2 = np.sum(np.square(x2), 1)

return x1p2.reshape((-1, 1)) + x2p2.reshape((1, -1)) - 2 * np.dot(x1, x2.T)</div>

def sinkhorn(w1, w2, M, reg, k):

r"""Sinkhorn algorithm with fixed number of iteration (autograd)"""

K = np.exp(-M / reg)

ui = np.ones((M.shape[0],))

vi = np.ones((M.shape[1],))

for i in range(k):

vi = w2 / (np.dot(K.T, ui) + 1e-50)

ui = w1 / (np.dot(K, vi) + 1e-50)

G = ui.reshape((M.shape[0], 1)) * K * vi.reshape((1, M.shape[1]))

return G</div>

def logsumexp(M, axis):

r"""Log-sum-exp reduction compatible with autograd (no numpy implementation)"""

amax = np.amax(M, axis=axis, keepdims=True)

return np.log(np.sum(np.exp(M - amax), axis=axis)) + np.squeeze(amax, axis=axis)</div>

def sinkhorn_log(w1, w2, M, reg, k):

r"""Sinkhorn algorithm in log-domain with fixed number of iteration (autograd)"""

Mr = -M / reg

ui = np.zeros((M.shape[0],))

vi = np.zeros((M.shape[1],))

log_w1 = np.log(w1)

log_w2 = np.log(w2)

for i in range(k):

vi = log_w2 - logsumexp(Mr + ui[:, None], 0)

ui = log_w1 - logsumexp(Mr + vi[None, :], 1)

G = np.exp(ui[:, None] + Mr + vi[None, :])

return G</div>

def split_classes(X, y):

r"""split samples in :math:`\mathbf{X}` by classes in :math:`\mathbf{y}`"""

lstsclass = np.unique(y)

return [X[y == i, :].astype(np.float32) for i in lstsclass]</div>

def fda(X, y, p=2, reg=1e-16):

r"""Fisher Discriminant Analysis

Parameters

----------

X : ndarray, shape (n, d)

Training samples.

y : ndarray, shape (n,)

Labels for training samples.

p : int, optional

Size of dimensionality reduction.

reg : float, optional

Regularization term >0 (ridge regularization)

Returns

-------

P : ndarray, shape (d, p)

Optimal transportation matrix for the given parameters

proj : callable

projection function including mean centering

"""

mx = np.mean(X)

X -= mx.reshape((1, -1))

# data split between classes

d = X.shape[1]

xc = split_classes(X, y)

nc = len(xc)

p = min(nc - 1, p)

Cw = 0

for x in xc:

Cw += np.cov(x, rowvar=False)

Cw /= nc

mxc = np.zeros((d, nc))

for i in range(nc):

mxc[:, i] = np.mean(xc[i])

mx0 = np.mean(mxc, 1)

Cb = 0

for i in range(nc):

Cb += (mxc[:, i] - mx0).reshape((-1, 1)) * (mxc[:, i] - mx0).reshape((1, -1))

w, V = linalg.eig(Cb, Cw + reg * np.eye(d))

idx = np.argsort(w.real)

Popt = V[:, idx[-p:]]

def proj(X):

return (X - mx.reshape((1, -1))).dot(Popt)

return Popt, proj</div>

def wda(

X,

y,

p=2,

reg=1,

k=10,

solver=None,

sinkhorn_method="sinkhorn",

maxiter=100,

verbose=0,

P0=None,

normalize=False,

):

r"""

Wasserstein Discriminant Analysis :ref:`[11] <references-wda>`

The function solves the following optimization problem:

.. math::

\mathbf{P} = \mathop{\arg \min}_\mathbf{P} \quad

\frac{\sum\limits_i W(P \mathbf{X}^i, P \mathbf{X}^i)}{\sum\limits_{i, j \neq i} W(P \mathbf{X}^i, P \mathbf{X}^j)}

where :

- :math:`P` is a linear projection operator in the Stiefel(`p`, `d`) manifold

- :math:`W` is entropic regularized Wasserstein distances

- :math:`\mathbf{X}^i` are samples in the dataset corresponding to class i

**Choosing a Sinkhorn solver**

By default and when using a regularization parameter that is not too small

the default sinkhorn solver should be enough. If you need to use a small

regularization to get sparse cost matrices, you should use the

:py:func:`ot.dr.sinkhorn_log` solver that will avoid numerical

errors, but can be slow in practice.

Parameters

----------

X : ndarray, shape (n, d)

Training samples.

y : ndarray, shape (n,)

Labels for training samples.

p : int, optional

Size of dimensionality reduction.

reg : float, optional

Regularization term >0 (entropic regularization)

solver : None | str, optional

None for steepest descent or 'TrustRegions' for trust regions algorithm

else should be a pymanopt.solvers

sinkhorn_method : str

method used for the Sinkhorn solver, either 'sinkhorn' or 'sinkhorn_log'

P0 : ndarray, shape (d, p)

Initial starting point for projection.

normalize : bool, optional

Normalize the Wasserstaiun distance by the average distance on P0 (default : False)

verbose : int, optional

Print information along iterations.

Returns

-------

P : ndarray, shape (d, p)

Optimal transportation matrix for the given parameters

proj : callable

Projection function including mean centering.

.. _references-wda:

References

----------

.. [11] Flamary, R., Cuturi, M., Courty, N., & Rakotomamonjy, A. (2016).

Wasserstein Discriminant Analysis. arXiv preprint arXiv:1608.08063.

""" # noqa

if sinkhorn_method.lower() == "sinkhorn":

sinkhorn_solver = sinkhorn

elif sinkhorn_method.lower() == "sinkhorn_log":

sinkhorn_solver = sinkhorn_log

else:

raise ValueError("Unknown Sinkhorn method '%s'." % sinkhorn_method)

mx = np.mean(X)

# data split between classes

d = X.shape[1]

# compute uniform weighs

wc = [np.ones((x.shape[0]), dtype=np.float32) / x.shape[0] for x in xc]

# pre-compute reg_c,c'

if P0 is not None and normalize:

regmean = np.zeros((len(xc), len(xc)))

for i, xi in enumerate(xc):

xi = np.dot(xi, P0)

for j, xj in enumerate(xc[i:]):

xj = np.dot(xj, P0)

M = dist(xi, xj)

regmean[i, j] = np.sum(M) / (len(xi) * len(xj))

else:

regmean = np.ones((len(xc), len(xc)))

manifold = pymanopt.manifolds.Stiefel(d, p)

@pymanopt.function.autograd(manifold)

def cost(P):

# wda loss

loss_b = 0

loss_w = 0

G = sinkhorn_solver(wc[i], wc[j + i], M, reg * regmean[i, j], k)

if j == 0:

loss_w += np.sum(G * M)

else:

loss_b += np.sum(G * M)

# loss inversed because minimization

return loss_w / loss_b

# declare manifold and problem

problem = pymanopt.Problem(manifold=manifold, cost=cost)

# declare solver and solve

if solver is None:

solver = pymanopt.optimizers.SteepestDescent(

max_iterations=maxiter, log_verbosity=verbose

)

elif solver in ["tr", "TrustRegions"]:

)

Popt = solver.run(problem, initial_point=P0)

def proj(X):

return Popt.point, proj</div>

def projection_robust_wasserstein(

X,

Y,

a,

b,

tau,

U0=None,

reg=0.1,

k=2,

stopThr=1e-3,

maxiter=100,

verbose=0,

random_state=None,

):

r"""

Projection Robust Wasserstein Distance :ref:`[32] <references-projection-robust-wasserstein>`

The function solves the following optimization problem:

.. math::

\max_{U \in St(d, k)} \ \min_{\pi \in \Pi(\mu,\nu)} \quad \sum_{i,j} \pi_{i,j}

\|U^T(\mathbf{x}_i - \mathbf{y}_j)\|^2 - \mathrm{reg} \cdot H(\pi)

- :math:`U` is a linear projection operator in the Stiefel(`d`, `k`) manifold

- :math:`H(\pi)` is entropy regularizer

- :math:`\mathbf{x}_i`, :math:`\mathbf{y}_j` are samples of measures :math:`\mu` and :math:`\nu` respectively

Parameters

----------

X : ndarray, shape (n, d)

Samples from measure :math:`\mu`

Y : ndarray, shape (n, d)

Samples from measure :math:`\nu`

a : ndarray, shape (n, )

weights for measure :math:`\mu`

b : ndarray, shape (n, )

weights for measure :math:`\nu`

tau : float

stepsize for Riemannian Gradient Descent

U0 : ndarray, shape (d, p)

Initial starting point for projection.

reg : float, optional

Regularization term >0 (entropic regularization)

k : int

Subspace dimension

stopThr : float, optional

Stop threshold on error (>0)

verbose : int, optional

Print information along iterations.

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

Determines random number generation for initial value of projection

operator when U0 is not given.

Returns

-------

pi : ndarray, shape (n, n)

Optimal transportation matrix for the given parameters

U : ndarray, shape (d, k)

Projection operator.

.. _references-projection-robust-wasserstein:

References

----------

.. [32] Huang, M. , Ma S. & Lai L. (2021).

A Riemannian Block Coordinate Descent Method for Computing

the Projection Robust Wasserstein Distance, ICML.

""" # noqa

# initialization

n, d = X.shape

m, d = Y.shape

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

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

u = np.ones(n) / n

v = np.ones(m) / m

ones = np.ones((n, m))

assert d > k

if U0 is None:

rng = check_random_state(random_state)

U = rng.randn(d, k)

else:

U = U0

def Vpi(X, Y, a, b, pi):

# Return the second order matrix of the displacements: sum_ij { (pi)_ij (X_i-Y_j)(X_i-Y_j)^T }.

A = X.T.dot(pi).dot(Y)

return (

X.T.dot(np.diag(a)).dot(X)

+ Y.T.dot(np.diag(np.sum(pi, 0))).dot(Y)

- A

- A.T

)

err = 1

iter = 0

while err > stopThr and iter < maxiter:

# Projected cost matrix

UUT = U.dot(U.T)

M = (

np.diag(np.diag(X.dot(UUT.dot(X.T)))).dot(ones)

+ ones.dot(np.diag(np.diag(Y.dot(UUT.dot(Y.T)))))

- 2 * X.dot(UUT.dot(Y.T))

)

A = np.empty(M.shape, dtype=M.dtype)

np.divide(M, -reg, out=A)

np.exp(A, out=A)

# Sinkhorn update

Ap = (1 / a).reshape(-1, 1) * A

AtransposeU = np.dot(A.T, u)

v = np.divide(b, AtransposeU)

u = 1.0 / np.dot(Ap, v)

pi = u.reshape((-1, 1)) * A * v.reshape((1, -1))

V = Vpi(X, Y, a, b, pi)

# Riemannian gradient descent

G = 2 / reg * V.dot(U)

GTU = G.T.dot(U)

xi = G - U.dot(GTU + GTU.T) / 2 # Riemannian gradient

U, _ = np.linalg.qr(U + tau * xi) # Retraction by QR decomposition

grad_norm = np.linalg.norm(xi)

err = max(reg * grad_norm, np.linalg.norm(np.sum(pi, 0) - b, 1))

f_val = np.trace(U.T.dot(V.dot(U)))

if verbose:

print("RBCD Iteration: ", iter, " error", err, "\t fval: ", f_val)

iter = iter + 1

return pi, U</div>

def ewca(

X,

U0=None,

reg=1,

k=2,

method="BCD",

sinkhorn_method="sinkhorn",

stopThr=1e-6,

maxiter=100,

maxiter_sink=1000,

maxiter_MM=10,

verbose=0,

):

r"""

Entropic Wasserstein Component Analysis :ref:`[52] <references-entropic-wasserstein-component_analysis>`.

The function solves the following optimization problem:

.. math::

\mathbf{U} = \mathop{\arg \min}_\mathbf{U} \quad

W(\mathbf{X}, \mathbf{U}\mathbf{U}^T \mathbf{X})

where :

- :math:`\mathbf{U}` is a matrix in the Stiefel(`p`, `d`) manifold

- :math:`W` is entropic regularized Wasserstein distances

- :math:`\mathbf{X}` are samples

Parameters

----------

X : ndarray, shape (n, d)

Samples from measure :math:`\mu`.

U0 : ndarray, shape (d, k), optional

Initial starting point for projection.

reg : float, optional

Regularization term >0 (entropic regularization).

k : int, optional

Subspace dimension.

method : str, optional

Eather 'BCD' or 'MM' (Block Coordinate Descent or Majorization-Minimization).

Prefer MM when d is large.

sinkhorn_method : str

Method used for the Sinkhorn solver, see :ref:`ot.bregman.sinkhorn` for more details.

stopThr : float, optional

Stop threshold on error (>0).

maxiter : int, optional

Maximum number of iterations of the BCD/MM.

maxiter_sink : int, optional

Maximum number of iterations of the Sinkhorn solver.

maxiter_MM : int, optional

Maximum number of iterations of the MM (only used when method='MM').

verbose : int, optional

Print information along iterations.

Returns

-------

pi : ndarray, shape (n, n)

Optimal transportation matrix for the given parameters.

U : ndarray, shape (d, k)

Matrix Stiefel manifold.

.. _references-entropic-wasserstein-component_analysis:

References

----------

.. [52] Collas, A., Vayer, T., Flamary, F., & Breloy, A. (2023).

Entropic Wasserstein Component Analysis.

""" # noqa

n, d = X.shape

X = X - X.mean(0)

if U0 is None:

pca_fitted = PCA(n_components=k).fit(X)

U = pca_fitted.components_.T

if method == "MM":

lambda_scm = pca_fitted.explained_variance_[0]

else:

U = U0

# marginals

u0 = (1.0 / n) * np.ones(n)

# print iterations

if verbose > 0:

print(

"{:4s}|{:13s}|{:12s}|{:12s}".format("It.", "Loss", "Crit.", "Thres.")

+ "\n"

+ "-" * 40

)

def compute_loss(M, pi, reg):

return np.sum(M * pi) + reg * np.sum(pi * (np.log(pi) - 1))

def grassmann_distance(U1, U2):

proj = U1.T @ U2

_, s, _ = np.linalg.svd(proj)

s[s > 1] = 1

s = np.arccos(s)

return np.linalg.norm(s)

# loop

it = 0

crit = np.inf

sinkhorn_warmstart = None

while (it < maxiter) and (crit > stopThr):

U_old = U

# Solve transport

M = dist_utils(X, (X @ U) @ U.T)

pi, log_sinkhorn = sinkhorn_bregman(

u0,

M,

reg,

numItermax=maxiter_sink,

method=sinkhorn_method,

warmstart=sinkhorn_warmstart,

warn=False,

log=True,

)

key_warmstart = "warmstart"

if key_warmstart in log_sinkhorn:

sinkhorn_warmstart = log_sinkhorn[key_warmstart]

if (pi >= 1e-300).all():

loss = compute_loss(M, pi, reg)

else:

loss = np.inf

# Solve PCA

pi_sym = (pi + pi.T) / 2

if method == "BCD":

# block coordinate descent

S = X.T @ (2 * pi_sym - (1.0 / n) * np.eye(n)) @ X

_, U = np.linalg.eigh(S)

U = U[:, ::-1][:, :k]

elif method == "MM":

# majorization-minimization

eig, _ = np.linalg.eigh(pi_sym)

lambda_pi = eig[0]

for _ in range(maxiter_MM):

X_proj = X @ U

X_T_X_proj = X.T @ X_proj

R = (1 / n) * X_T_X_proj

alpha = 1 - 2 * n * lambda_pi

if alpha > 0:

R = alpha * (R - lambda_scm * U)

else:

R = alpha * R

R = R - (2 * X.T @ (pi_sym @ X_proj)) + (2 * lambda_pi * X_T_X_proj)

else:

raise ValueError(f"Unknown method '{method}', use 'BCD' or 'MM'.")

# stop or not

it += 1

crit = grassmann_distance(U_old, U)

# print

if verbose > 0:

print("{:4d}|{:8e}|{:8e}|{:8e}".format(it, loss, crit, stopThr))

return pi, U</div>

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