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<!DOCTYPE html>

<head>

<title>ot.sliced — POT Python Optimal Transport 0.9.6 documentation</title>

</head>

POT Python Optimal Transport

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

<li class="breadcrumb-item active">ot.sliced</li>

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

"""

Sliced OT Distances

"""

# Author: Adrien Corenflos <adrien.corenflos@aalto.fi>

# Nicolas Courty <ncourty@irisa.fr>

# Rémi Flamary <remi.flamary@polytechnique.edu>

#

# License: MIT License

import numpy as np

from .backend import get_backend, NumpyBackend

from .utils import list_to_array, get_coordinate_circle

from .lp import (

wasserstein_circle,

semidiscrete_wasserstein2_unif_circle,

linear_circular_ot,

)

def get_random_projections(d, n_projections, seed=None, backend=None, type_as=None):

r"""

Generates n_projections samples from the uniform on the unit sphere of dimension :math:`d-1`: :math:`\mathcal{U}(\mathcal{S}^{d-1})`

Parameters

----------

d : int

dimension of the space

n_projections : int

number of samples requested

seed: int or RandomState, optional

Seed used for numpy random number generator

backend:

Backend to use for random generation

Returns

-------

out: ndarray, shape (d, n_projections)

The uniform unit vectors on the sphere

Examples

--------

>>> n_projections = 100

>>> d = 5

>>> projs = get_random_projections(d, n_projections)

>>> np.allclose(np.sum(np.square(projs), 0), 1.) # doctest: +NORMALIZE_WHITESPACE

True

"""

if backend is None:

nx = NumpyBackend()

else:

nx = backend

if isinstance(seed, np.random.RandomState) and str(nx) == "numpy":

projections = seed.randn(d, n_projections)

else:

if seed is not None:

nx.seed(seed)

projections = nx.randn(d, n_projections, type_as=type_as)

projections = projections / nx.sqrt(nx.sum(projections**2, 0, keepdims=True))

return projections</div>

def sliced_wasserstein_distance(

X_s,

X_t,

a=None,

b=None,

n_projections=50,

p=2,

projections=None,

seed=None,

log=False,

):

r"""

Computes a Monte-Carlo approximation of the p-Sliced Wasserstein distance

.. math::

\mathcal{SWD}_p(\mu, \nu) = \underset{\theta \sim \mathcal{U}(\mathbb{S}^{d-1})}{\mathbb{E}}\left(\mathcal{W}_p^p(\theta_\# \mu, \theta_\# \nu)\right)^{\frac{1}{p}}

where :

- :math:`\theta_\# \mu` stands for the pushforwards of the projection :math:`X \in \mathbb{R}^d \mapsto \langle \theta, X \rangle`

Parameters

----------

X_s : ndarray, shape (n_samples_a, dim)

samples in the source domain

X_t : ndarray, shape (n_samples_b, dim)

samples in the target domain

a : ndarray, shape (n_samples_a,), optional

samples weights in the source domain

b : ndarray, shape (n_samples_b,), optional

samples weights in the target domain

n_projections : int, optional

Number of projections used for the Monte-Carlo approximation

p: float, optional

Power p used for computing the sliced Wasserstein

projections: shape (dim, n_projections), optional

Projection matrix (n_projections and seed are not used in this case)

seed: int or RandomState or None, optional

Seed used for random number generator

log: bool, optional

if True, sliced_wasserstein_distance returns the projections used and their associated EMD.

Returns

-------

cost: float

Sliced Wasserstein Cost

log : dict, optional

log dictionary return only if log==True in parameters

Examples

--------

>>> n_samples_a = 20

>>> X = np.random.normal(0., 1., (n_samples_a, 5))

>>> sliced_wasserstein_distance(X, X, seed=0) # doctest: +NORMALIZE_WHITESPACE

0.0

References

----------

.. [31] Bonneel, Nicolas, et al. "Sliced and radon wasserstein barycenters of measures." Journal of Mathematical Imaging and Vision 51.1 (2015): 22-45

"""

from .lp import wasserstein_1d

X_s, X_t = list_to_array(X_s, X_t)

if a is not None and b is not None and projections is None:

nx = get_backend(X_s, X_t, a, b)

elif a is not None and b is not None and projections is not None:

elif a is None and b is None and projections is not None:

else:

n = X_s.shape[0]

m = X_t.shape[0]

if X_s.shape[1] != X_t.shape[1]:

raise ValueError(

"X_s and X_t must have the same number of dimensions {} and {} respectively given".format(

X_s.shape[1], X_t.shape[1]

)

if a is None:

a = nx.full(n, 1 / n, type_as=X_s)

if b is None:

b = nx.full(m, 1 / m, type_as=X_s)

d = X_s.shape[1]

if projections is None:

projections = get_random_projections(

d, n_projections, seed, backend=nx, type_as=X_s

)

else:

n_projections = projections.shape[1]

X_s_projections = nx.dot(X_s, projections)

X_t_projections = nx.dot(X_t, projections)

projected_emd = wasserstein_1d(X_s_projections, X_t_projections, a, b, p=p)

res = (nx.sum(projected_emd) / n_projections) ** (1.0 / p)

if log:

return res, {"projections": projections, "projected_emds": projected_emd}

return res</div>

def max_sliced_wasserstein_distance(

X_s,

X_t,

a=None,

b=None,

n_projections=50,

p=2,

projections=None,

seed=None,

log=False,

):

r"""

Computes a Monte-Carlo approximation of the max p-Sliced Wasserstein distance

.. math::

\mathcal{Max-SWD}_p(\mu, \nu) = \underset{\theta _in

\mathcal{U}(\mathbb{S}^{d-1})}{\max} [\mathcal{W}_p^p(\theta_\#

\mu, \theta_\# \nu)]^{\frac{1}{p}}

where :

- :math:`\theta_\# \mu` stands for the pushforwards of the projection :math:`\mathbb{R}^d \ni X \mapsto \langle \theta, X \rangle`

Parameters

----------

X_s : ndarray, shape (n_samples_a, dim)

samples in the source domain

X_t : ndarray, shape (n_samples_b, dim)

samples in the target domain

a : ndarray, shape (n_samples_a,), optional

samples weights in the source domain

b : ndarray, shape (n_samples_b,), optional

samples weights in the target domain

n_projections : int, optional

Number of projections used for the Monte-Carlo approximation

p: float, optional =

Power p used for computing the sliced Wasserstein

projections: shape (dim, n_projections), optional

Projection matrix (n_projections and seed are not used in this case)

seed: int or RandomState or None, optional

Seed used for random number generator

log: bool, optional

if True, sliced_wasserstein_distance returns the projections used and their associated EMD.

Returns

-------

cost: float

Sliced Wasserstein Cost

log : dict, optional

log dictionary return only if log==True in parameters

Examples

--------

>>> n_samples_a = 20

>>> X = np.random.normal(0., 1., (n_samples_a, 5))

>>> sliced_wasserstein_distance(X, X, seed=0) # doctest: +NORMALIZE_WHITESPACE

0.0

References

----------

.. [35] Deshpande, I., Hu, Y. T., Sun, R., Pyrros, A., Siddiqui, N., Koyejo, S., ... & Schwing, A. G. (2019). Max-sliced wasserstein distance and its use for gans. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. 10648-10656).

"""

from .lp import wasserstein_1d

else:

raise ValueError(

)

if a is None:

if b is None:

if projections is None:

projections = get_random_projections(

)

res = nx.max(projected_emd) ** (1.0 / p)

if log:

return res</div>

def get_projections_sphere(d, n_projections, seed=None, backend=None, type_as=None):

r"""

Generates n_projections samples from the uniform distribution on the Stiefel manifold of dimension :math:`d\times 2`: :math:`\mathbb{V}_{d,2}=\{X \in \mathbb{R}^{d\times 2}, X^TX=I_2\}`

Parameters

----------

d : int

dimension of the space

n_projections : int

number of samples requested

seed: int or RandomState, optional

Seed used for numpy random number generator

backend:

Backend to use for random generation

type_as: optional

Type to use for random generation

Returns

-------

out: ndarray, shape (n_projections, d, 2)

Examples

--------

>>> n_projections = 100

>>> d = 5

>>> projs = get_projections_sphere(d, n_projections)

>>> np.allclose(np.einsum("nij, nik -> njk", projs, projs), np.eye(2)) # doctest: +NORMALIZE_WHITESPACE

True

"""

if backend is None:

nx = NumpyBackend()

else:

nx = backend

Z = seed.randn(n_projections, d, 2)

else:

if seed is not None:

nx.seed(seed)

Z = nx.randn(n_projections, d, 2, type_as=type_as)

projections, _ = nx.qr(Z)

return projections</div>

def projection_sphere_to_circle(

x, n_projections=50, projections=None, seed=None, backend=None

):

r"""

Projection of :math:`x\in S^{d-1}` on circles using coordinates on [0,1[.

To get the projection on the circle, we use the following formula:

.. math::

P^U(x) = \frac{U^Tx}{\|U^Tx\|_2}

where :math:`U` is a random matrix sampled from the uniform distribution on the Stiefel manifold of dimension :math:`d\times 2`: :math:`\mathbb{V}_{d,2}=\{X \in \mathbb{R}^{d\times 2}, X^TX=I_2\}`

and :math:`x` is a point on the sphere. Then, we apply the function get_coordinate_circle to get the coordinates on :math:`[0,1[`.

Parameters

----------

x : ndarray, shape (n_samples, dim)

samples on the sphere

n_projections : int, optional

Number of projections used for the Monte-Carlo approximation

projections: shape (n_projections, dim, 2), optional

Projection matrix (n_projections and seed are not used in this case)

seed: int or RandomState or None, optional

Seed used for random number generator

backend:

Backend to use for random generation

Returns

-------

Xp_coords: ndarray, shape (n_projections, n_samples)

Coordinates of the projections on the circle

"""

if backend is None:

nx = get_backend(x)

else:

nx = backend

n, d = x.shape

if projections is None:

projections = get_projections_sphere(

d, n_projections, seed=seed, backend=nx, type_as=x

)

# Projection on S^1

# Projection on plane

Xp = nx.einsum("ikj, lk -> ilj", projections, x)

# Projection on sphere

Xp = Xp / nx.sqrt(nx.sum(Xp**2, -1, keepdims=True))

# Get coordinates on [0,1[

Xp_coords = nx.reshape(

get_coordinate_circle(nx.reshape(Xp, (-1, 2))), (n_projections, n)

)

return Xp_coords, projections</div>

def sliced_wasserstein_sphere(

X_s,

X_t,

a=None,

b=None,

n_projections=50,

p=2,

projections=None,

seed=None,

log=False,

):

r"""

Compute the spherical sliced-Wasserstein discrepancy.

.. math::

SSW_p(\mu,\nu) = \left(\int_{\mathbb{V}_{d,2}} W_p^p(P^U_\#\mu, P^U_\#\nu)\ \mathrm{d}\sigma(U)\right)^{\frac{1}{p}}

where:

- :math:`P^U_\# \mu` stands for the pushforwards of the projection :math:`\forall x\in S^{d-1},\ P^U(x) = \frac{U^Tx}{\|U^Tx\|_2}`

The function runs on backend but tensorflow and jax are not supported.

Parameters

----------

X_s: ndarray, shape (n_samples_a, dim)

Samples in the source domain

X_t: ndarray, shape (n_samples_b, dim)

Samples in the target domain

a : ndarray, shape (n_samples_a,), optional

samples weights in the source domain

b : ndarray, shape (n_samples_b,), optional

samples weights in the target domain

n_projections : int, optional

Number of projections used for the Monte-Carlo approximation

p: float, optional (default=2)

Power p used for computing the spherical sliced Wasserstein

projections: shape (n_projections, dim, 2), optional

Projection matrix (n_projections and seed are not used in this case)

seed: int or RandomState or None, optional

Seed used for random number generator

log: bool, optional

if True, sliced_wasserstein_sphere returns the projections used and their associated EMD.

Returns

-------

cost: float

Spherical Sliced Wasserstein Cost

log: dict, optional

log dictionary return only if log==True in parameters

Examples

--------

>>> n_samples_a = 20

>>> X = np.random.normal(0., 1., (n_samples_a, 5))

>>> X = X / np.sqrt(np.sum(X**2, -1, keepdims=True))

>>> sliced_wasserstein_sphere(X, X, seed=0) # doctest: +NORMALIZE_WHITESPACE

0.0

References

----------

.. [46] Bonet, C., Berg, P., Courty, N., Septier, F., Drumetz, L., & Pham, M. T. (2023). Spherical sliced-wasserstein. International Conference on Learning Representations.

"""

else:

raise ValueError(

)

if nx.any(nx.abs(nx.sum(X_s**2, axis=-1) - 1) > 10 ** (-4)):

raise ValueError("X_s is not on the sphere.")

if nx.any(nx.abs(nx.sum(X_t**2, axis=-1) - 1) > 10 ** (-4)):

raise ValueError("X_t is not on the sphere.")

if projections is None:

projections = get_projections_sphere(

)

Xps_coords, _ = projection_sphere_to_circle(

X_s, n_projections=n_projections, projections=projections, seed=seed, backend=nx

)

Xpt_coords, _ = projection_sphere_to_circle(

X_t, n_projections=n_projections, projections=projections, seed=seed, backend=nx

)

projected_emd = wasserstein_circle(

Xps_coords.T, Xpt_coords.T, u_weights=a, v_weights=b, p=p

)

res = nx.mean(projected_emd) ** (1 / p)

if log:

return res</div>

def sliced_wasserstein_sphere_unif(

X_s, a=None, n_projections=50, projections=None, seed=None, log=False

):

r"""Compute the 2-spherical sliced wasserstein w.r.t. a uniform distribution.

.. math::

SSW_2(\mu_n, \nu)

where

- :math:`\mu_n=\sum_{i=1}^n \alpha_i \delta_{x_i}`

- :math:`\nu=\mathrm{Unif}(S^{d-1})`

Parameters

----------

X_s: ndarray, shape (n_samples_a, dim)

Samples in the source domain

a : ndarray, shape (n_samples_a,), optional

samples weights in the source domain

n_projections : int, optional

Number of projections used for the Monte-Carlo approximation

projections: shape (n_projections, dim, 2), optional

Projection matrix (n_projections and seed are not used in this case)

seed: int or RandomState or None, optional

Seed used for random number generator

log: bool, optional

if True, sliced_wasserstein_distance returns the projections used and their associated EMD.

Returns

-------

cost: float

Spherical Sliced Wasserstein Cost

log: dict, optional

log dictionary return only if log==True in parameters

Examples

---------

>>> np.random.seed(42)

>>> x0 = np.random.randn(500,3)

>>> x0 = x0 / np.sqrt(np.sum(x0**2, -1, keepdims=True))

>>> ssw = sliced_wasserstein_sphere_unif(x0, seed=42)

>>> np.allclose(sliced_wasserstein_sphere_unif(x0, seed=42), 0.01734, atol=1e-3)

True

References:

-----------

.. [46] Bonet, C., Berg, P., Courty, N., Septier, F., Drumetz, L., & Pham, M. T. (2023). Spherical sliced-wasserstein. International Conference on Learning Representations.

"""

if a is not None:

else:

nx = get_backend(X_s)

raise ValueError("X_s is not on the sphere.")

if projections is None:

projections = get_projections_sphere(

)

Xps_coords, _ = projection_sphere_to_circle(

)

projected_emd = semidiscrete_wasserstein2_unif_circle(Xps_coords.T, u_weights=a)

if log:

return res</div>

def linear_sliced_wasserstein_sphere(

X_s,

X_t=None,

a=None,

b=None,

n_projections=50,

projections=None,

seed=None,

log=False,

):

r"""Computes the linear spherical sliced wasserstein distance from :ref:`[79] <references-lssot>`.

General loss returned:

.. math::

\mathrm{LSSOT}_2(\mu, \nu) = \left(\int_{\mathbb{V}_{d,2}} \mathrm{LCOT}_2^2(P^U_\#\mu, P^U_\#\nu)\ \mathrm{d}\sigma(U)\right)^{\frac12},

where :math:`\mu,\nu\in\mathcal{P}(S^{d-1})` are two probability measures on the sphere, :math:`\mathrm{LCOT}_2` is the linear circular optimal transport distance,

and :math:`P^U_\# \mu` stands for the pushforwards of the projection :math:`\forall x\in S^{d-1},\ P^U(x) = \frac{U^Tx}{\|U^Tx\|_2}`.

Parameters

----------

X_s: ndarray, shape (n_samples_a, dim)

Samples in the source domain

X_t: ndarray, shape (n_samples_b, dim), optional

Samples in the target domain. If None, computes the distance against the uniform distribution on the sphere.

a : ndarray, shape (n_samples_a,), optional

samples weights in the source domain

b : ndarray, shape (n_samples_b,), optional

samples weights in the target domain

n_projections : int, optional

Number of projections used for the Monte-Carlo approximation

projections: shape (n_projections, dim, 2), optional

Projection matrix (n_projections and seed are not used in this case)

seed: int or RandomState or None, optional

Seed used for random number generator

log: bool, optional

if True, linear_sliced_wasserstein_sphere returns the projections used and their associated LCOT.

Returns

-------

cost: float

Linear Spherical Sliced Wasserstein Cost

log: dict, optional

log dictionary return only if log==True in parameters

Examples

---------

>>> n_samples_a = 20

>>> X = np.random.normal(0., 1., (n_samples_a, 5))

>>> X = X / np.sqrt(np.sum(X**2, -1, keepdims=True))

>>> linear_sliced_wasserstein_sphere(X, X, seed=0) # doctest: +NORMALIZE_WHITESPACE

0.0

.. _references-lssot:

References

----------

.. [79] Liu, X., Bai, Y., Martín, R. D., Shi, K., Shahbazi, A., Landman, B. A., Chang, C., & Kolouri, S. (2025). Linear Spherical Sliced Optimal Transport: A Fast Metric for Comparing Spherical Data. International Conference on Learning Representations.

"""

else:

raise ValueError(

)

raise ValueError("X_s is not on the sphere.")

if nx.any(nx.abs(nx.sum(X_t**2, axis=-1) - 1) > 10 ** (-4)):

raise ValueError("X_t is not on the sphere.")

if projections is None:

projections = get_projections_sphere(

)

Xps_coords, _ = projection_sphere_to_circle(

)

if X_t is not None:

Xpt_coords, _ = projection_sphere_to_circle(

X_t,

n_projections=n_projections,

projections=projections,

seed=seed,

backend=nx,

)

projected_lcot = linear_circular_ot(

)

res = nx.mean(projected_lcot) ** (1 / 2)

if log:

return res</div>

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