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<!DOCTYPE html>

<head>

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

</head>

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

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

"""

Optimal transport for Gaussian distributions

"""

# Author: Theo Gnassounou <theo.gnassounou@inria.fr>

# Remi Flamary <remi.flamary@polytehnique.edu>

#

# License: MIT License

import warnings

import numpy as np

from .backend import get_backend

from .utils import dots, is_all_finite, list_to_array, exp_bures

def bures_wasserstein_mapping(ms, mt, Cs, Ct, log=False):

r"""Return OT linear operator between samples.

The function estimates the optimal linear operator that aligns the two

empirical distributions. This is equivalent to estimating the closed

form mapping between two Gaussian distributions :math:`\mathcal{N}(\mu_s,\Sigma_s)`

and :math:`\mathcal{N}(\mu_t,\Sigma_t)` as proposed in

:ref:`[1] <references-OT-mapping-linear>` and discussed in remark 2.29 in

:ref:`[2] <references-OT-mapping-linear>`.

The linear operator from source to target :math:`M`

.. math::

M(\mathbf{x})= \mathbf{A} \mathbf{x} + \mathbf{b}

where :

.. math::

\mathbf{A} &= \Sigma_s^{-1/2} \left(\Sigma_s^{1/2}\Sigma_t\Sigma_s^{1/2} \right)^{1/2}

\Sigma_s^{-1/2}

\mathbf{b} &= \mu_t - \mathbf{A} \mu_s

Parameters

----------

ms : array-like (d,)

mean of the source distribution

mt : array-like (d,)

mean of the target distribution

Cs : array-like (d,d)

covariance of the source distribution

Ct : array-like (d,d)

covariance of the target distribution

log : bool, optional

record log if True

Returns

-------

A : (d, d) array-like

Linear operator

b : (1, d) array-like

bias

log : dict

log dictionary return only if log==True in parameters

.. _references-OT-mapping-linear:

References

----------

.. [1] Knott, M. and Smith, C. S. "On the optimal mapping of

distributions", Journal of Optimization Theory and Applications

Vol 43, 1984

.. [2] Peyré, G., & Cuturi, M. (2017). "Computational Optimal

Transport", 2018.

"""

ms, mt, Cs, Ct = list_to_array(ms, mt, Cs, Ct)

Cs12 = nx.sqrtm(Cs)

Cs12inv = nx.inv(Cs12)

M0 = nx.sqrtm(dots(Cs12, Ct, Cs12))

A = dots(Cs12inv, M0, Cs12inv)

b = mt - nx.dot(ms, A)

if log:

log = {}

log["Cs12"] = Cs12

log["Cs12inv"] = Cs12inv

return A, b, log

else:

return A, b</div>

def bures_wasserstein_mapping_hd(ms, mt, Us, Ut, ls, lt, sigma2_s, sigma2_t, log=False):

r"""Return OT linear operator between HD Gaussian distritutions.

The function estimates the optimal linear operator that aligns the two

HD Gaussian distributions :math:`\mathcal{N}(\mu_s, U_s, l_s, \sigma_s^2, d_s)`

and :math:`\mathcal{N}(\mu_t, U_t, l_t, \sigma_t^2, d_t)` as proposed in

:ref:`[88] <references-OT-mapping-linear>`, Th. 2.9

.

The linear operator from source to target :math:`M`

.. math::

M(\mathbf{x})= \mathbf{A} \mathbf{x} + \mathbf{b}

where :

.. math::

\mathbf{A} &= \Sigma_s^{-1/2} \left(\Sigma_s^{1/2}\Sigma_t\Sigma_s^{1/2} \right)^{1/2}

\Sigma_s^{-1/2} \\

\Sigma_s^{1/2} &=\sigma_s I_p + U_s C_s U_s^T \\

C_s &=\text{diag}(\sqrt{l_{s1} + \sigma_s^2} - \sigma_s, \dots, \sqrt{l_{sd_s} + \sigma_s^2} - \sigma_s) \\

\Sigma_s^{-1/2} &= \frac{1}{\sigma_s} (I_p - U_s D_s U_s^T ) \\

D_s &= \text{diag}((\sqrt{l_{s1} + \sigma_s^2} - \sigma_s)/\sqrt{l_{s1} + \sigma_s^2}, \dots, (\sqrt{l_{sd_s} + \sigma_s^2} - \sigma_s)/\sqrt{l_{sd_s} + \sigma_s^2}) \\

\Sigma_t &= U_t \text{diag}(l_t) U_t^T + \sigma_t^2 I_p \\

\mathbf{b} &= \mu_t - \mathbf{A} \mu_s

Parameters

----------

ms : array-like (p,)

mean of the source distribution

mt : array-like (p,)

mean of the target distribution

Us : array-like (p,ds)

orthogonal matrix spanning the principal subspace of the source distribution

Ut : array-like (p,dt)

orthogonal matrix spanning the principal subspace of the target distribution

ls : array-like (ds,)

the variances associated with the principal sub-axes for the source distribution

lt : array-like (dt,)

the variances associated with the principal sub-axes for the target distribution

sigma_s^2 : array-like (1,)

the residual variance of the source distribution

sigma_t^2 : array-like (1,)

the residual variance of the target distribution

log : bool, optional

record log if True

Returns

-------

A : (d, d) array-like

Linear operator

b : (1, d) array-like

bias

log : dict

log dictionary return only if log==True in parameters

.. _references-OT-mapping-linear:

References

----------

.. [1] Knott, M. and Smith, C. S. "On the optimal mapping of

distributions", Journal of Optimization Theory and Applications

Vol 43, 1984

.. [2] Peyré, G., & Cuturi, M. (2017). "Computational Optimal

Transport", 2018.

.. [88] Bouveyron, C. & Corneli, M. ("Scaling Optimal Transport to High-Dimensional Gaussian Distributions")

"""

nx = get_backend(ms, mt, Us, Ut, ls, lt, sigma2_s, sigma2_t)

p = Us.shape[0]

# source

Cs = nx.diag(nx.sqrt(ls + sigma2_s) - nx.sqrt(sigma2_s))

Ss_sq = dots(Us, Cs, Us.T) + nx.sqrt(sigma2_s) * nx.eye(p)

Ds = nx.diag((nx.sqrt(ls + sigma2_s) - nx.sqrt(sigma2_s)) / nx.sqrt(ls + sigma2_s))

Ss_sqinv = (1 / nx.sqrt(sigma2_s)) * (nx.eye(p) - dots(Us, Ds, Us.T))

# destination

St = dots(Ut, nx.diag(lt), Ut.T) + sigma2_t * nx.eye(p)

M0 = nx.sqrtm(dots(Ss_sq, St, Ss_sq))

A = dots(Ss_sqinv, M0, Ss_sqinv)

if log:

log = {}

log["Ss_sq"] = Ss_sq

log["Ss_sqinv"] = Ss_sqinv

return A, b, log

else:

return A, b</div>

def empirical_bures_wasserstein_mapping(

xs, xt, reg=1e-6, ws=None, wt=None, bias=True, log=False

):

r"""Return OT linear operator between samples.

The function estimates the optimal linear operator that aligns the two

empirical distributions. This is equivalent to estimating the closed

form mapping between two Gaussian distributions :math:`\mathcal{N}(\mu_s,\Sigma_s)`

and :math:`\mathcal{N}(\mu_t,\Sigma_t)` as proposed in

:ref:`[1] <references-OT-mapping-linear>` and discussed in remark 2.29 in

:ref:`[2] <references-OT-mapping-linear>`.

The linear operator from source to target :math:`M`

.. math::

M(\mathbf{x})= \mathbf{A} \mathbf{x} + \mathbf{b}

where :

.. math::

\mathbf{A} &= \Sigma_s^{-1/2} \left(\Sigma_s^{1/2}\Sigma_t\Sigma_s^{1/2} \right)^{1/2}

\Sigma_s^{-1/2}

\mathbf{b} &= \mu_t - \mathbf{A} \mu_s

Parameters

----------

xs : array-like (ns,d)

samples in the source domain

xt : array-like (nt,d)

samples in the target domain

reg : float,optional

regularization added to the diagonals of covariances (>0)

ws : array-like (ns,1), optional

weights for the source samples

wt : array-like (nt,1), optional

weights for the target samples

bias: boolean, optional

estimate bias :math:`\mathbf{b}` else :math:`\mathbf{b} = 0` (default:True)

log : bool, optional

record log if True

Returns

-------

A : (d, d) array-like

Linear operator

b : (1, d) array-like

bias

log : dict

log dictionary return only if log==True in parameters

.. _references-OT-mapping-linear:

References

----------

.. [1] Knott, M. and Smith, C. S. "On the optimal mapping of

distributions", Journal of Optimization Theory and Applications

Vol 43, 1984

.. [2] Peyré, G., & Cuturi, M. (2017). "Computational Optimal

Transport", 2018.

"""

xs, xt = list_to_array(xs, xt)

nx = get_backend(xs, xt)

is_input_finite = is_all_finite(xs, xt)

d = xs.shape[1]

if ws is None:

ws = nx.ones((xs.shape[0], 1), type_as=xs) / xs.shape[0]

if wt is None:

wt = nx.ones((xt.shape[0], 1), type_as=xt) / xt.shape[0]

if bias:

mxs = nx.dot(ws.T, xs) / nx.sum(ws)

mxt = nx.dot(wt.T, xt) / nx.sum(wt)

xs = xs - mxs

xt = xt - mxt

else:

mxs = nx.zeros((1, d), type_as=xs)

mxt = nx.zeros((1, d), type_as=xs)

Cs = nx.dot((xs * ws).T, xs) / nx.sum(ws) + reg * nx.eye(d, type_as=xs)

Ct = nx.dot((xt * wt).T, xt) / nx.sum(wt) + reg * nx.eye(d, type_as=xt)

if log:

A, b, log = bures_wasserstein_mapping(mxs, mxt, Cs, Ct, log=log)

else:

A, b = bures_wasserstein_mapping(mxs, mxt, Cs, Ct)

if is_input_finite and not is_all_finite(A, b):

warnings.warn(

"Numerical errors were encountered in ot.gaussian.empirical_bures_wasserstein_mapping. "

"Consider increasing the regularization parameter `reg`."

)

if log:

log["Cs"] = Cs

log["Ct"] = Ct

return A, b, log

else:

return A, b</div>

def empirical_bures_wasserstein_mapping_hd(

xs, xt, d_intrinsic, reg=0.0, ws=None, wt=None, bias=True, log=False

):

r"""Return OT HD linear operator between samples.

The function estimates the optimal linear HD operator that aligns the two

empirical distributions. This is equivalent to estimating the closed

form mapping between two HD Gaussian distributions :math:`\mathcal{N}(\mu_s, U_s, l_s, \sigma_s^2, d_s)`

and :math:`\mathcal{N}(\mu_t, U_t, l_t, \sigma_t^2, d_t)` as proposed in

:ref:`[88] <references-empirical-bures-wasserstein-mapping-hd>`, Th. 2.9.

The linear operator from source to target :math:`M`

.. math::

M(\mathbf{x})= \mathbf{A} \mathbf{x} + \mathbf{b}

where :

.. math::

\mathbf{A} &= \Sigma_s^{-1/2} \left(\Sigma_s^{1/2}\Sigma_t\Sigma_s^{1/2} \right)^{1/2}

\Sigma_s^{-1/2} \\

\Sigma_s^{1/2} &=\sigma_s I_p + U_s C_s U_s^T \\

C_s &=\text{diag}(\sqrt{l_{s1} + \sigma_s^2} - \sigma_s, \dots, \sqrt{l_{sd_s} + \sigma_s^2} - \sigma_s) \\

\Sigma_s^{-1/2} &= \frac{1}{\sigma_s} (I_p - U_s D_s U_s^T ) \\

D_s &= \text{diag}((\sqrt{l_{s1} + \sigma_s^2} - \sigma_s)/\sqrt{l_{s1} + \sigma_s^2}, \dots, (\sqrt{l_{sd_s} + \sigma_s^2} - \sigma_s)/\sqrt{l_{sd_s} + \sigma_s^2}) \\

\Sigma_t &= U_t \text{diag}(l_t) U_t^T + \sigma_t^2 I_p \\

\mathbf{b} &= \mu_t - \mathbf{A} \mu_s

Assuming that the source and destination data samples have been generated from HD Gaussian distributions, the probabilistic PCA estimators proposed in :ref: `[89] <references-empirical-bures-wasserstein-distance-hd>` are used to estimate the model parameters and plugged into the above formulas.

Parameters

----------

xs : array-like (ns,p)

samples in the source domain

xt : array-like (nt,p)

samples in the target domain

d_instrinsic : int or list of two ints

the intrinsic dimensions of the source and destination distribution, respectively. If d_intrinsic is int it is assumed the intrinsic dimension is the same

reg : float,optional

regularization added to the diagonals of covariances (null by default)

ws : array-like (ns,1), optional

weights for the source samples

wt : array-like (nt,1), optional

weights for the target samples

bias: boolean, optional

estimate bias :math:`\mathbf{b}` else :math:`\mathbf{b} = 0` (default:True)

log : bool, optional

record log if True

Returns

-------

A : (p, p) array-like

Linear operator

b : (1, p) array-like

bias

log : dict

log dictionary return only if log==True in parameters

.. _references-empirical-bures-wasserstein-hd:

References

----------

.. [88] Bouveyron, C. & Corneli, M. ("Scaling Optimal Transport to High-Dimensional Gaussian Distributions")

[89] Tipping, M.E. & Bishop, C.M, ("Probabilistic Principal Component Analysis")

"""

(ds, dt) = (

(d_intrinsic, d_intrinsic) if isinstance(d_intrinsic, int) else d_intrinsic

)

p = xs.shape[1]

if ws is None:

if wt is None:

if bias:

xs = xs - mxs

xt = xt - mxt

else:

mxs = nx.zeros((1, p), type_as=xs)

mxt = nx.zeros((1, p), type_as=xs)

Cs = nx.dot((xs * ws).T, xs) / nx.sum(ws) + reg * nx.eye(p, type_as=xs)

Ct = nx.dot((xt * wt).T, xt) / nx.sum(wt) + reg * nx.eye(p, type_as=xt)

eigs = nx.eigh(Cs)

a_s = eigs[0][-ds:]

sgm2_s = (nx.trace(Cs) - nx.sum(a_s)) / (p - ds)

Qs = eigs[1]

Us = Qs[:, -ds:]

ls = a_s - sgm2_s

eigt = nx.eigh(Ct)

a_t = eigt[0][-dt:]

sgm2_t = (nx.trace(Ct) - nx.sum(a_t)) / (p - dt)

Qt = eigt[1]

Ut = Qt[:, -dt:]

lt = a_t - sgm2_t

sgm2_s = nx.maximum(nx.unsqueeze(sgm2_s, 0), reg)

sgm2_t = nx.maximum(nx.unsqueeze(sgm2_t, 0), reg)

if log:

mxs, mxt, Us, Ut, ls, lt, sgm2_s, sgm2_t, log=log

)

else:

A, b = bures_wasserstein_mapping_hd(mxs, mxt, Us, Ut, ls, lt, sgm2_s, sgm2_t)

warnings.warn(

"Numerical errors were encountered in ot.gaussian.empirical_bures_wasserstein_mapping_hd. "

"Consider increasing the regularization parameter `reg` or reducing the intrinsic dimensions ds/dt."

)

if log:

log["Us"] = Us

log["Ut"] = Ut

log["ls"] = ls

log["lt"] = lt

log["sigma2_s"] = sgm2_s

log["sigma2_t"] = sgm2_t

return A, b, log

else:

return A, b</div>

def bures_distance(Cs, Ct, paired=False, log=False, nx=None):

r"""Return Bures distance.

The function computes the Bures distance between :math:`\mu_s=\mathcal{N}(0,\Sigma_s)` and :math:`\mu_t=\mathcal{N}(0,\Sigma_t)`,

given by (see e.g. Remark 2.31 :ref:`[15] <references-bures-wasserstein-distance>`):

.. math::

\mathbf{B}(\Sigma_s, \Sigma_t)^{2} = \text{Tr}\left(\Sigma_s + \Sigma_t - 2 \sqrt{\Sigma_s^{1/2}\Sigma_t\Sigma_s^{1/2}} \right)

Parameters

----------

Cs : array-like (d,d) or (n,d,d)

covariance of the source distribution

Ct : array-like (d,d) or (m,d,d)

covariance of the target distribution

paired: bool, optional

if True and n==m, return the paired distances and crossed distance otherwise

log : bool, optional

record log if True

nx : module, optional

The numerical backend module to use. If not provided, the backend will

be fetched from the input matrices `Cs, Ct`.

Returns

-------

W : float if Cs and Cd of shape (d,d), array-like (n,m) if Cs of shape (n,d,d) and Ct of shape (m,d,d), array-like (n,) if Cs and Ct of shape (n, d, d) and paired is True

Bures Wasserstein distance

log : dict

log dictionary return only if log==True in parameters

.. _references-bures-wasserstein-distance:

References

----------

.. [15] Peyré, G., & Cuturi, M. (2019). Computational optimal transport: With applications to data science.

Foundations and Trends® in Machine Learning, 11(5-6), 355-607.

"""

Cs, Ct = list_to_array(Cs, Ct)

if nx is None:

nx = get_backend(Cs, Ct)

assert Cs.shape[-1] == Ct.shape[-1], "All Gaussian must have the same dimension"

if len(Cs.shape) == 2 and len(Ct.shape) == 2:

# Return float

bw2 = nx.trace(Cs + Ct - 2 * nx.sqrtm(dots(Cs12, Ct, Cs12)))

else:

assert (

len(Cs.shape) == 3 and len(Ct.shape) == 3

), "Both Cs and Ct should be batched"

if paired and len(Cs) == len(Ct):

# Return shape (n,)

M = nx.einsum("nij, njk, nkl -> nil", Cs12, Ct, Cs12)

else:

# Return shape (n,m)

M = nx.einsum("nij, mjk, nkl -> nmil", Cs12, Ct, Cs12)

bw2 = nx.trace(Cs[:, None] + Ct[None] - 2 * nx.sqrtm(M))

W = nx.sqrt(nx.maximum(bw2, 0))

if log:

log = {}

log["Cs12"] = Cs12

return W, log

else:

return W</div>

def bures_wasserstein_distance(ms, mt, Cs, Ct, paired=False, log=False):

r"""Return Bures Wasserstein distance between samples.

The function computes the Bures-Wasserstein distance between :math:`\mu_s=\mathcal{N}(m_s,\Sigma_s)` and :math:`\mu_t=\mathcal{N}(m_t,\Sigma_t)`,

as discussed in remark 2.31 :ref:`[15] <references-bures-wasserstein-distance>`.

.. math::

\mathcal{W}(\mu_s, \mu_t)_2^2= \left\lVert \mathbf{m}_s - \mathbf{m}_t \right\rVert^2 + \mathcal{B}(\Sigma_s, \Sigma_t)^{2}

where :

.. math::

\mathbf{B}(\Sigma_s, \Sigma_t)^{2} = \text{Tr}\left(\Sigma_s + \Sigma_t - 2 \sqrt{\Sigma_s^{1/2}\Sigma_t\Sigma_s^{1/2}} \right)

Parameters

----------

ms : array-like (d,) or (n,d)

mean of the source distribution

mt : array-like (d,) or (m,d)

mean of the target distribution

Cs : array-like (d,d) or (n,d,d)

covariance of the source distribution

Ct : array-like (d,d) or (m,d,d)

covariance of the target distribution

paired: bool, optional

if True and n==m, return the paired distances and crossed distance otherwise

log : bool, optional

record log if True

Returns

-------

W : float if ms and md of shape (d,), array-like (n,m) if ms of shape (n,d) and mt of shape (m,d), array-like (n,) if ms and mt of shape (n,d) and paired is True

Bures Wasserstein distance

log : dict

log dictionary return only if log==True in parameters

.. _references-bures-wasserstein-distance:

References

----------

.. [15] Peyré, G., & Cuturi, M. (2019). Computational optimal transport: With applications to data science.

Foundations and Trends® in Machine Learning, 11(5-6), 355-607.

"""

assert (

ms.shape[0] == Cs.shape[0]

), "Source Gaussians has different amount of components"

assert (

mt.shape[0] == Ct.shape[0]

), "Target Gaussians has different amount of components"

assert (

ms.shape[-1] == mt.shape[-1] == Cs.shape[-1] == Ct.shape[-1]

), "All Gaussian must have the same dimension"

if log:

bw, log_dict = bures_distance(Cs, Ct, paired=paired, log=log, nx=nx)

Cs12 = log_dict["Cs12"]

else:

bw = bures_distance(Cs, Ct, paired=paired, nx=nx)

if len(ms.shape) == 1 and len(mt.shape) == 1:

# Return float

else:

assert (

len(ms.shape) == 2 and len(mt.shape) == 2

), "Both ms and mt should be batched"

if paired and len(ms.shape) == len(mt.shape):

# Return shape (n,)

squared_dist_m = nx.norm(ms - mt, axis=-1) ** 2

else:

# Return shape (n,m)

squared_dist_m = nx.norm(ms[:, None] - mt[None], axis=-1) ** 2

W = nx.sqrt(nx.maximum(squared_dist_m + bw**2, 0))

if log:

log = {}

log["Cs12"] = Cs12

return W, log

else:

return W</div>

def bures_wasserstein_distance_hd(

ms, mt, Us, Ut, ls, lt, sigma2_s, sigma2_t, log=False

):

r"""Return the 2-Wassestein distance between HD Gaussian distritutions.

The function estimates the 2-Wasserstein distance between the two

HD Gaussian distributions :math:`\mathcal{N}(\mu_s, U_s, l_s, \sigma_s^2, d_s)`

and :math:`\mathcal{N}(\mu_t, U_t, l_t, \sigma_t^2, d_t)` as proposed in

:ref:`[88] <references-bures-wasserstein-distance-hd>`, Prop. 2.3

.. math::

\mathcal{W}(\mu_s, \mu_t)_2^2= \left\lVert \mathbf{m}_s - \mathbf{m}_t \right\rVert^2 + \text{Tr}(\Lambda_s) + \text{Tr}(\Lambda_t) + p(\sigma_s^2 + \sigma_t^2) - 2\text{Tr}((\Sigma_s^{(1/2)}\Sigma_t \Sigma_s^{1/2})^{1/2})

where :

.. math::

\Lambda_s &= \text{diag}(ls) \\

\Lambda_t &= \text{diag}(lt) \\

\Sigma_t &= U_t \Lambda_t U_t^T + \sigma_t^2 I_p \\

\Sigma_s^{1/2} &=\sigma_s I_p + U_s C_s U_s^T \\

C_s &=\text{diag}(\sqrt{l_{s1} + \sigma_s^2} - \sigma_s, \dots, \sqrt{l_{sd_s} + \sigma_s^2} - \sigma_s) \\

Parameters

----------

ms : array-like (p,)

mean of the source distribution

mt : array-like (p,)

mean of the target distribution

Us : array-like (p,ds)

orthogonal matrix spanning the principal subspace of the source distribution

Ut : array-like (p,dt)

orthogonal matrix spanning the principal subspace of the target distribution

ls : array-like (ds,)

the variances associated with the principal sub-axes for the source distribution

lt : array-like (dt,)

the variances associated with the principal sub-axes for the target distribution

sigma_s^2 : array-like (1,)

the residual variance of the source distribution

sigma_t^2 : array-like(1,)

the residual variance of the target distribution

log : bool, optional

record log if True

Returns

-------

W : float

Bures Wasserstein distance

.. _references-bures-wasserstein-distance-hd:

References

----------

.. [1] Knott, M. and Smith, C. S. "On the optimal mapping of

distributions", Journal of Optimization Theory and Applications

Vol 43, 1984

.. [2] Peyré, G., & Cuturi, M. (2017). "Computational Optimal

Transport", 2018.

.. [88] Bouveyron, C. & Corneli, M. ("Scaling Optimal Transport to High-Dimensional Gaussian Distributions")

"""

# source

# destination

A = dots(Ss_sq, St, Ss_sq)

W2 = (

nx.sum((ms - mt) ** 2)

+ nx.sum(ls)

+ nx.sum(lt)

+ p * (sigma2_s + sigma2_t)

- 2 * nx.trace(nx.sqrtm(A))

)

W = nx.sqrt(W2)

if log:

log = {}

log["Ss_sq"] = Ss_sq

log["St"] = St

return W, log

else:

return W</div>

def empirical_bures_wasserstein_distance(

):

r"""Return Bures Wasserstein distance from mean and covariance of distribution.

The function estimates the Bures-Wasserstein distance between two

empirical distributions source :math:`\mu_s` and target :math:`\mu_t`,

discussed in remark 2.31 :ref:`[1] <references-bures-wasserstein-distance>`.

The Bures Wasserstein distance between source and target distribution :math:`\mathcal{W}`

.. math::

\mathcal{W}(\mu_s, \mu_t)_2^2= \left\lVert \mathbf{m}_s - \mathbf{m}_t \right\rVert^2 + \mathcal{B}(\Sigma_s, \Sigma_t)^{2}

where :

.. math::

\mathbf{B}(\Sigma_s, \Sigma_t)^{2} = \text{Tr}\left(\Sigma_s + \Sigma_t - 2 \sqrt{\Sigma_s^{1/2}\Sigma_t\Sigma_s^{1/2}} \right)

Parameters

----------

xs : array-like (ns,d)

samples in the source domain

xt : array-like (nt,d)

samples in the target domain

reg : float,optional

regularization added to the diagonals of covariances (>0)

ws : array-like (ns), optional

weights for the source samples

wt : array-like (ns), optional

weights for the target samples

bias: boolean, optional

estimate bias :math:`\mathbf{b}` else :math:`\mathbf{b} = 0` (default:True)

log : bool, optional

record log if True

Returns

-------

W : float

Bures Wasserstein distance

log : dict

log dictionary return only if log==True in parameters

.. _references-bures-wasserstein-distance:

References

----------

.. [1] Peyré, G., & Cuturi, M. (2017). "Computational Optimal

Transport", 2018.

"""

if ws is None:

if wt is None:

if bias:

xs = xs - mxs

xt = xt - mxt

else:

if log:

W, log = bures_wasserstein_distance(mxs[0], mxt[0], Cs, Ct, log=log)

log["Cs"] = Cs

log["Ct"] = Ct

return W, log

else:

W = bures_wasserstein_distance(mxs[0], mxt[0], Cs, Ct)

return W</div>

def empirical_bures_wasserstein_distance_hd(

):

r"""Return 2D Wasserstein between high-dimensional (HD) Gaussian distributions.

The function estimates the 2-Wasserstein distance between the two

HD Gaussian distributions :math:`\mathcal{N}(\mu_s, U_s, l_s, \sigma_s^2, d_s)`

and :math:`\mathcal{N}(\mu_t, U_t, l_t, \sigma_t^2, d_t)` as proposed in

:ref:`[88] <references-empirical-bures-wasserstein-distance-hd>`, Prop. 2.3

.. math::

where :

.. math::

\Lambda_s &= \text{diag}(ls) \\

\Lambda_t &= \text{diag}(lt) \\

\Sigma_t &= U_t \Lambda_t U_t^T + \sigma_t^2 I_p \\

\Sigma_s^{1/2} &=\sigma_s I_p + U_s C_s U_s^T \\

C_s &=\text{diag}(\sqrt{l_{s1} + \sigma_s^2} - \sigma_s, \dots, \sqrt{l_{sd_s} + \sigma_s^2} - \sigma_s) \\

Parameters

----------

xs : array-like (ns,p)

samples in the source domain

xt : array-like (nt,p)

samples in the target domain

d_instrinsic : int or list of two ints

the intrinsic dimensions of the source and destination distribution, respectively. If d_intrinsic is int it is assumed the intrinsic dimension is the same

reg : float,optional

regularization added to the diagonals of covariances (null by default)

ws : array-like (ns,1), optional

weights for the source samples

wt : array-like (nt,1), optional

weights for the target samples

bias: boolean, optional

estimate bias :math:`\mathbf{b}` else :math:`\mathbf{b} = 0` (default:True)

log : bool, optional

record log if True

Returns

-------

A : (p, p) array-like

Linear operator

b : (1, p) array-like

bias

log : dict

log dictionary return only if log==True in parameters

.. _references-empirical-bures-wasserstein-distance-hd:

References

----------

.. [88] Bouveyron, C. & Corneli, M. ("Scaling Optimal Transport to High-Dimensional Gaussian Distributions")

[89] Tipping, M.E. & Bishop, C.M, ("Probabilistic Principal Component Analysis")

"""

(ds, dt) = (

)

if ws is None:

if wt is None:

if bias:

xs = xs - mxs

xt = xt - mxt

else:

mxs = nx.zeros((1, p), type_as=xs)

mxt = nx.zeros((1, p), type_as=xs)

Cs = nx.dot((xs * ws).T, xs) / nx.sum(ws) + reg * nx.eye(p, type_as=xs)

Ct = nx.dot((xt * wt).T, xt) / nx.sum(wt) + reg * nx.eye(p, type_as=xt)

Qs = eigs[1]

Us = Qs[:, -ds:]

ls = a_s - sgm2_s

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