Gaussians and covariances

Gaussian Wasserstein / Bures

gaussian_wasserstein_distances

all-pairspairedcondensedsingledistanceGaussians

W_2^2=\|\mu-\nu\|_2^2+\operatorname{tr}\!\left(\Sigma+\Lambda-2(\Sigma^{1/2}\Lambda\Sigma^{1/2})^{1/2}\right)

Means and finite symmetric positive-semidefinite covariances. Default returns W2; squared=True returns the expression shown. Singular covariance boundaries use a finite derivative continuation.

Shapes and Modes

Input	Layout
One sample	`(mean[D], covariance[D,D])`
Collection X	`(means[...,N,D], covariances[...,N,D,D])`
Collection Y	`(means[...,M,D], covariances[...,M,D,D])`

Mode	Selector	Output
all-pairs	`compare(X,Y,metric=...)`	`[...,N,M]`
paired	`compare(X,Y,metric=...,mode="paired")`	`[...,N]`; equal sample counts
condensed	`compare(X,metric=...,mode="condensed")`	`[...,N*(N-1)//2]`; strict upper triangle
single	`compare(x,y,metric=...,mode="single")`	`[...]`; leading batches only

Leading batch dimensions broadcast. See mode and shape conventions.

Entry Points

Named Python entry	Function
All-pairs	`gaussian_wasserstein_distances`
Paired	`paired_gaussian_wasserstein_distances`
Condensed	`condensed_gaussian_wasserstein_distances`

The Python and C++ compare selectors cover every mode shown above. cdist / pairwise_distances and paired_distances select this comparison; pdist / condensed_distances select its condensed mode. single_comparison is shorthand for single mode.

Catalog aliases: gaussian_wasserstein.

Accepted metric options: squared. Omitted options keep the metric defaults; unused options are rejected.

Example

Choose Python or C++ in the top bar; the setting applies across this guide.

Python

import torch
import torchpairwise as tp

X = (torch.rand(2, 2), torch.eye(2).repeat(2, 1, 1))
Y = (torch.rand(3, 2), 2 * torch.eye(2).repeat(3, 1, 1))
# (means [N,D], covariances [N,D,D])

metric = "gaussian_wasserstein_distances"
all_pairs = tp.compare(X, Y, metric=metric)  # [2,3]
paired_Y = tuple(t[:2] for t in Y)
paired = tp.compare(X, paired_Y, metric=metric, mode="paired")  # [2]
condensed = tp.compare(X, metric=metric, mode="condensed")      # [1]
x, y = tuple(t[0] for t in X), tuple(t[0] for t in Y)
single = tp.compare(x, y, metric=metric, mode="single")  # scalar

C++ / LibTorch

#include <torchpairwise.h>

namespace tp = torchpairwise::ops;
tp::ComparisonCollection X{at::rand({2, 2}), at::eye(2).repeat({2, 1, 1})};
tp::ComparisonCollection Y{at::rand({3, 2}), 2 * at::eye(2).repeat({3, 1, 1})};

const std::string metric = "gaussian_wasserstein_distances";
tp::ComparisonOptions options;
auto all_pairs = tp::compare(X, Y, metric, "all_pairs", options);  // [2,3]
tp::ComparisonCollection paired_Y{Y[0].narrow(0, 0, 2), Y[1].narrow(0, 0, 2)};
auto paired = tp::compare(X, paired_Y, metric, "paired", options);  // [2]
// [1]
auto condensed = tp::compare(X, c10::nullopt, metric, "condensed", options);
tp::ComparisonCollection x{X[0][0], X[1][0]};
tp::ComparisonCollection y{Y[0][0], Y[1][0]};
auto single = tp::compare(x, y, metric, "single", options);  // scalar

Streamed Evaluation

tp.lazy(x, y, name) binds this distance for reductions, selection and products without retaining its full generated comparison matrix. Native source factories depend on the sample family; see streamed operators and C++ use.

Numerical Contract

Floating CPU/CUDA inputs use matching float32 / float64, with lower precision promoted under autocast where supported. First derivatives follow the metric’s smooth and boundary conventions. See the detailed conventions and precision and deployment.