"""
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
[docs]
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
[docs]
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:
nx = get_backend(X_s, X_t, a, b, projections)
elif a is None and b is None and projections is not None:
nx = get_backend(X_s, X_t, projections)
else:
nx = get_backend(X_s, X_t)
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
[docs]
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
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:
nx = get_backend(X_s, X_t, a, b, projections)
elif a is None and b is None and projections is not None:
nx = get_backend(X_s, X_t, projections)
else:
nx = get_backend(X_s, X_t)
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
)
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.max(projected_emd) ** (1.0 / p)
if log:
return res, {"projections": projections, "projected_emds": projected_emd}
return res
[docs]
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.
"""
if a is not None and b is not None:
nx = get_backend(X_s, X_t, a, b)
else:
nx = get_backend(X_s, X_t)
n, d = X_s.shape
m, _ = X_t.shape
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 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:
# Uniforms and independent samples on the Stiefel manifold V_{d,2}
if isinstance(seed, np.random.RandomState) and str(nx) == "numpy":
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=X_s)
projections, _ = nx.qr(Z)
else:
n_projections = projections.shape[0]
# Projection on S^1
# Projection on plane
Xps = nx.einsum("ikj, lk -> ilj", projections, X_s)
Xpt = nx.einsum("ikj, lk -> ilj", projections, X_t)
# Projection on sphere
Xps = Xps / nx.sqrt(nx.sum(Xps**2, -1, keepdims=True))
Xpt = Xpt / nx.sqrt(nx.sum(Xpt**2, -1, keepdims=True))
# Get coordinates on [0,1[
Xps_coords = nx.reshape(
get_coordinate_circle(nx.reshape(Xps, (-1, 2))), (n_projections, n)
)
Xpt_coords = nx.reshape(
get_coordinate_circle(nx.reshape(Xpt, (-1, 2))), (n_projections, m)
)
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, {"projections": projections, "projected_emds": projected_emd}
return res
[docs]
def sliced_wasserstein_sphere_unif(X_s, a=None, n_projections=50, 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^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
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:
nx = get_backend(X_s, a)
else:
nx = get_backend(X_s)
n, d = X_s.shape
if nx.any(nx.abs(nx.sum(X_s**2, axis=-1) - 1) > 10 ** (-4)):
raise ValueError("X_s is not on the sphere.")
# Uniforms and independent samples on the Stiefel manifold V_{d,2}
if isinstance(seed, np.random.RandomState) and str(nx) == "numpy":
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=X_s)
projections, _ = nx.qr(Z)
# Projection on S^1
# Projection on plane
Xps = nx.einsum("ikj, lk -> ilj", projections, X_s)
# Projection on sphere
Xps = Xps / nx.sqrt(nx.sum(Xps**2, -1, keepdims=True))
# Get coordinates on [0,1[
Xps_coords = nx.reshape(
get_coordinate_circle(nx.reshape(Xps, (-1, 2))), (n_projections, n)
)
projected_emd = semidiscrete_wasserstein2_unif_circle(Xps_coords.T, u_weights=a)
res = nx.mean(projected_emd) ** (1 / 2)
if log:
return res, {"projections": projections, "projected_emds": projected_emd}
return res
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