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<!DOCTYPE html>

<head>

<title>ot.dr — POT Python Optimal Transport 0.9.8.dev documentation</title>

</head>

POT Python Optimal Transport

</a>

</form>

</div>

</div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu">

<ul>

<li class="toctree-l1"><a class="reference internal" href="../../index.html">POT: Python Optimal Transport</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../auto_examples/plot_quickstart_guide.html">Quickstart Guide</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../auto_examples/index.html">Examples gallery</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../user_guide.html">User guide</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../all.html">API and modules</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../releases.html">Releases</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../contributors.html">Contributors</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../contributing.html">Contributing to POT</a></li>

<li class="toctree-l1"><a class="reference internal" href="../../code_of_conduct.html">Code of conduct</a></li>

</ul>

</div>

</nav>

<a href="../../index.html">POT Python Optimal Transport</a>

</nav>

<li class="breadcrumb-item"><a href="../index.html">Module code</a></li>

</li>

</ul>

<hr/>

</div>

<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

# ot.dr offers solvers with different dependencies. Each is imported optionally

# so that, for instance, the PyTorch WDA solver works on an installation with no

# autograd or pymanopt. Functions raise an ImportError naming what they need.

try:

import autograd.numpy as np

HAS_AUTOGRAD = True

except ImportError: # pragma: no cover - depends on the installation

import numpy as np

HAS_AUTOGRAD = False

try:

import pymanopt

import pymanopt.manifolds

import pymanopt.optimizers

HAS_PYMANOPT = True

except ImportError: # pragma: no cover - depends on the installation

HAS_PYMANOPT = False

try:

import torch

HAS_TORCH = True

except ImportError: # pragma: no cover - depends on the installation

HAS_TORCH = False

try:

from sklearn.decomposition import PCA

HAS_SKLEARN = True

except ImportError: # pragma: no cover - depends on the installation

HAS_SKLEARN = False

from .backend import get_backend

from .bregman import sinkhorn as sinkhorn_bregman

from .utils import dist as dist_utils, check_random_state

def _require(condition, function, dependencies):

"""Check requirement and raise error if not available."""

if not condition:

raise ImportError(

f"Missing dependency for ot.dr.{function}. Requires {dependencies}. "

"You can install with 'pip install POT[dr]', or "

"'conda install autograd pymanopt scikit-learn'"

)

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 _stiefel_projection(P, G, nx):

r"""Project a euclidean gradient onto the tangent space of the Stiefel manifold.

Parameters

----------

P : array-like, shape (d, p)

Current point, with orthonormal columns.

G : array-like, shape (d, p)

Euclidean gradient at :math:`\mathbf{P}`.

nx : backend

Backend to use for the computation.

Returns

-------

G_tangent : array-like, shape (d, p)

Component of :math:`\mathbf{G}` tangent to Stiefel at :math:`\mathbf{P}`,

that is :math:`\mathbf{G} - \mathbf{P}\,\mathrm{sym}(\mathbf{P}^\top \mathbf{G})`.

"""

W = nx.dot(P.T, G)

return G - nx.dot(P, 0.5 * (W + W.T))

def _stiefel_retraction(P, X, nx):

r"""Retract :math:`\mathbf{P} + \mathbf{X}` onto the Stiefel manifold by QR.

The sign of each column is fixed so that the diagonal of :math:`\mathbf{R}`

is non-negative, which makes the retraction a continuous map.

Parameters

----------

P : array-like, shape (d, p)

Current point, with orthonormal columns.

X : array-like, shape (d, p)

Tangent step to apply.

nx : backend

Backend to use for the computation.

Returns

-------

P_next : array-like, shape (d, p)

Point on Stiefel with orthonormal columns.

"""

Q, R = nx.qr(P + X)

d = nx.diag(R)

return Q * nx.sign(nx.sign(d) + 0.5)

def _wda_cost(P, xc, wc, regmean, reg, k, sinkhorn_method, nx):

r"""Wasserstein Discriminant Analysis objective at a projection :math:`\mathbf{P}`.

Computes the ratio of the within-class to the between-class entropic

transport cost of the projected samples, which is the quantity `wda`

minimizes. The Sinkhorn solver is run for exactly `k` iterations rather

than to a tolerance, so the objective is a fixed-depth, differentiable

function of :math:`\mathbf{P}`.

Parameters

----------

P : array-like, shape (d, p)

Projection, with orthonormal columns.

xc : list of array-like

Samples split by class, each of shape (n_i, d).

wc : list of array-like

Uniform weights for each class, each of shape (n_i,).

regmean : array-like, shape (n_classes, n_classes)

Per-pair scaling of `reg`, all ones unless `normalize` was requested.

reg : float

Entropic regularization term > 0.

k : int

Number of Sinkhorn iterations.

sinkhorn_method : str

Either 'sinkhorn' or 'sinkhorn_log', passed to :py:func:`ot.sinkhorn`.

nx : backend

Backend to use for the computation.

Returns

-------

loss : float or array-like scalar

Within-class cost divided by between-class cost.

"""

loss_b, loss_w = 0.0, 0.0

for i, xi in enumerate(xc):

xi = nx.dot(xi, P)

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

xj = nx.dot(xj, P)

M = dist_utils(xi, xj)

G = sinkhorn_bregman(

wc[i],

wc[j + i],

M,

reg * regmean[i, j],

method=sinkhorn_method,

numItermax=k,

stopThr=0.0,

warn=False,

)

term = nx.sum(G * M)

if j == 0:

loss_w = loss_w + term

else:

loss_b = loss_b + term

if float(nx.to_numpy(loss_b)) == 0.0:

raise ValueError(

"The between-class transport cost underflowed to zero, so the WDA "

"objective is undefined. reg is too small for the scale of the "

"data: exp(-M/reg) underflows. Increase reg, or use "

"sinkhorn_method='sinkhorn_log'."

)

return loss_w / loss_b

def _wda_torch(

X,

y,

p,

reg,

k,

sinkhorn_method,

maxiter,

verbose,

P0,

normalize,

random_state,

step_growth,

step_shrink,

max_backtracks,

gtol,

):

r"""Solve WDA with PyTorch autodiff and Riemannian gradient descent.

Follows the same scheme as pymanopt's ``SteepestDescent`` with its default

``BackTrackingLineSearcher``: project the euclidean gradient onto the

tangent space, retract by QR, and backtrack from a step proportional to the

last accepted one. Both solvers therefore target the same optimum.

Parameters

----------

X : torch.Tensor, shape (n, d)

Mean-centred training samples.

y : torch.Tensor, shape (n,)

Integer class codes.

p : int

Size of the dimensionality reduction.

reg : float

Entropic regularization term > 0.

k : int

Number of Sinkhorn iterations per objective evaluation.

sinkhorn_method : str

Either 'sinkhorn' or 'sinkhorn_log'.

maxiter : int

Maximum number of descent iterations.

verbose : int

Print the objective and gradient norm at each iteration if non-zero.

P0 : torch.Tensor, shape (d, p) or None

Starting point. Drawn at random when None.

normalize : bool

Scale `reg` per class pair by the mean projected distance at `P0`.

random_state : int, RandomState or None

Seeds the starting point when `P0` is None.

step_growth : float

Factor applied to the last accepted step to start the line search.

step_shrink : float

Factor applied on each backtracking step.

max_backtracks : int

Maximum number of backtracking steps before the solver stops.

gtol : float

Stop once the tangent gradient norm falls below this value.

Returns

-------

P : torch.Tensor, shape (d, p)

Optimal projection.

"""

nx = get_backend(X)

dtype, device = X.dtype, X.device

labels = torch.unique(y)

xc = [X[y == c] for c in labels]

wc = [

torch.full((x.shape[0],), 1.0 / x.shape[0], dtype=dtype, device=device)

for x in xc

]

d = X.shape[1]

nc = len(xc)

if not 1 <= p <= d:

raise ValueError(f"Need d >= p >= 1. Values supplied were d = {d} and p = {p}")

if P0 is None:

rng = check_random_state(random_state)

P = torch.linalg.qr(torch.tensor(rng.randn(d, p), dtype=dtype, device=device))[

0

]

else:

P = P0.clone().to(dtype=dtype, device=device)

regmean = torch.ones((nc, nc), dtype=dtype, device=device)

if P0 is not None and normalize:

with torch.no_grad():

xi = xi @ P

xj = xj @ P

regmean[i, j] = torch.sum(dist_utils(xi, xj)) / (

xi.shape[0] * xj.shape[0]

)

def value(Q):

with torch.no_grad():

return _wda_cost(Q, xc, wc, regmean, reg, k, sinkhorn_method, nx)

f = value(P)

step = 1.0

for it in range(maxiter):

Q = P.detach().requires_grad_(True)

v = _wda_cost(Q, xc, wc, regmean, reg, k, sinkhorn_method, nx)

(g,) = torch.autograd.grad(v, Q)

direction = -_stiefel_projection(P, g, nx)

gnorm = float(nx.norm(direction))

if verbose:

print(f"{it + 1:<6d} {float(v.detach()):+.16e} {gnorm:.8e}")

if gnorm <= gtol:

break

step = step_growth * step

improved = False

for _ in range(max_backtracks):

Pn = _stiefel_retraction(P, step * direction, nx)

fn = value(Pn)

if fn < f:

improved = True

break

step = step_shrink * step

if not improved:

break

P, f = Pn, fn

return P.detach()

def _wda_torch_entry(

X,

y,

p,

reg,

k,

sinkhorn_method,

maxiter,

verbose,

P0,

normalize,

random_state,

step_growth,

step_shrink,

max_backtracks,

gtol,

):

r"""Prepare inputs for the torch solver and wrap its output.

A torch tensor keeps its dtype and device throughout. Any other array type

is promoted to float64 on the CPU, matching what the autograd solver

returns, and the projection comes back as a numpy array.

Parameters

----------

X : array-like, shape (n, d)

Training samples, not modified.

y : array-like, shape (n,)

Labels, of any type numpy can take the unique values of.

Other parameters are those of :py:func:`ot.dr.wda`.

Returns

-------

P : array-like, shape (d, p)

Optimal projection, of the same array type as `X`.

proj : callable

Projection function including mean centering.

"""

was_numpy = not torch.is_tensor(X)

if was_numpy:

Xt = torch.as_tensor(np.asarray(X, dtype=np.float64))

else:

Xt = X if torch.is_floating_point(X) else X.to(torch.float64)

# torch cannot hold arbitrary label types, so index classes by position the

# way split_classes does for numpy

if torch.is_tensor(y):

yt = y.to(Xt.device)

else:

yt = torch.as_tensor(

np.unique(np.asarray(y), return_inverse=True)[1], device=Xt.device

)

if P0 is None:

P0t = None

else:

P0t = (P0 if torch.is_tensor(P0) else torch.as_tensor(np.asarray(P0))).to(

dtype=Xt.dtype, device=Xt.device

)

mx = Xt.mean(dim=0)

Xc = Xt - mx.reshape((1, -1))

Popt = _wda_torch(

Xc,

yt,

p,

reg,

k,

sinkhorn_method,

maxiter,

verbose,

P0t,

normalize,

random_state,

step_growth,

step_shrink,

max_backtracks,

gtol,

)

if was_numpy:

Pn = Popt.detach().cpu().numpy()

mxn = mx.detach().cpu().numpy()

def proj(Z):

return (Z - mxn.reshape((1, -1))).dot(Pn)

return Pn, proj

def proj(Z):

return (Z - mx.reshape((1, -1))) @ Popt

return Popt, proj

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, axis=0)

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], axis=0)

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_autograd(

X, y, p, reg, k, solver, sinkhorn_method, maxiter, verbose, P0, normalize

):

r"""Solve WDA with autograd and a pymanopt optimizer.

Parameters

----------

X : ndarray, shape (n, d)

Training samples, not modified.

y : ndarray, shape (n,)

Labels for training samples.

solver : None | str | pymanopt.optimizers

``None`` or ``'autograd'`` selects ``SteepestDescent``, ``'tr'`` or

``'TrustRegions'`` selects ``TrustRegions``, and any other value is

used as a pymanopt optimizer instance.

Other parameters are those of :py:func:`ot.dr.wda`.

Returns

-------

P : ndarray, shape (d, p)

Optimal projection.

proj : callable

Projection function including mean centering.

"""

if solver == "autograd":

solver = None

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)

# 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'

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

xi = np.dot(xi, P0)

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

def wda(

X,

y,

p=2,

reg=1,

k=10,

solver=None,

sinkhorn_method="sinkhorn",

maxiter=100,

verbose=0,

P0=None,

normalize=False,

random_state=None,

step_growth=2.0,

step_shrink=0.5,

max_backtracks=40,

gtol=1e-12,

):

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 | pymanopt.optimizers, optional

Chooses both the autodiff framework and the optimizer.

- `None` or `'autograd'` (default): autograd and pymanopt

`SteepestDescent`.

- `'TrustRegions'` (or `'tr'`): autograd and pymanopt `TrustRegions`.

- a `pymanopt.optimizers` instance: autograd with that optimizer.

- `'torch'`: PyTorch autodiff with Riemannian gradient descent and a QR

retraction. Requires only `torch`, so it works on installations

without autograd or pymanopt, and accepts torch tensors directly,

keeping their device and dtype.

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.

random_state : int, RandomState instance or None, optional

Seeds the random starting point when `P0` is None. Only used by

`solver='torch'`; the pymanopt solvers draw their own starting point.

step_growth : float, optional

Factor applied to the last accepted step size to start the next line

search (default 2.0). Only used by `solver='torch'`.

step_shrink : float, optional

Factor applied to the step size on each backtracking step

(default 0.5). Only used by `solver='torch'`.

max_backtracks : int, optional

Maximum number of backtracking steps per iteration before the solver

stops (default 40). Only used by `solver='torch'`.

gtol : float, optional

Stop once the norm of the tangent gradient falls below this value

(default 1e-12). Only used by `solver='torch'`.

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 HAS_TORCH and torch.is_tensor(y):

n_classes = int(torch.unique(y).numel())

else:

n_classes = np.unique(np.asarray(y)).size

if n_classes < 2:

raise ValueError(

f"WDA needs at least two classes, got {n_classes}: with a single "

"class the between-class transport cost is zero and the objective "

"is undefined."

)

if solver == "torch":

_require(HAS_TORCH, "wda(solver='torch')", "torch")

return _wda_torch_entry(

X,

y,

p,

reg,

k,

sinkhorn_method,

maxiter,

verbose,

P0,

normalize,

random_state,

step_growth,

step_shrink,

max_backtracks,

gtol,

)

_require(

HAS_AUTOGRAD and HAS_PYMANOPT,

"wda(solver='autograd')",

"autograd and pymanopt",

)

return _wda_autograd(

)</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>

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