Sinkhorn Divergence

Debiased entropy-regularized optimal transport, interpolating between Wasserstein distance and MMD.

sinkhorn_divergence(samples_p, samples_q, *, epsilon=0.01, p=2, max_iter=100, tol=1e-06)

Compute the debiased Sinkhorn divergence between two sample sets.

.. math::

S_\varepsilon(P, Q) = OT_\varepsilon(P, Q)
    - \frac{1}{2}\bigl(OT_\varepsilon(P, P) + OT_\varepsilon(Q, Q)\bigr)

where :math:OT_\varepsilon is the entropy-regularized optimal transport cost with regularization parameter :math:\varepsilon.

Parameters:

Name	Type	Description	Default
samples_p	ndarray	Samples from distribution P. Shape (n,) for 1D or (n, d) for d-dimensional data.	required
samples_q	ndarray	Samples from distribution Q. Shape (m,) for 1D or (m, d) for d-dimensional data.	required
epsilon	float	Entropic regularization parameter. Smaller values approximate the Wasserstein distance more closely but require more iterations. Default is 0.01.	0.01
p	int	Exponent for the ground cost: c(x, y) = ||x - y||^p. Default is 2.	2
max_iter	int	Maximum number of Sinkhorn iterations. Default is 100.	100
tol	float	Convergence tolerance. Default is 1e-6.	1e-06

Returns:

Type	Description
float	The debiased Sinkhorn divergence, non-negative.

Notes

The debiasing correction subtracts the self-transport costs to ensure that :math:S_\varepsilon(P, P) = 0. Without debiasing, the regularized OT cost is always positive even for identical distributions.

Examples:

>>> import numpy as np
>>> rng = np.random.default_rng(42)
>>> p_samples = rng.normal(0, 1, 200)
>>> q_samples = rng.normal(2, 1, 200)
>>> sinkhorn_divergence(p_samples, q_samples) > 0
True