The Gromov–Wasserstein distance—a generalization of the usual Wasserstein distance—permits comparing probability measures defined on possibly different metric spaces. Recently, this notion of distance has found several applications in Data Science and in Machine Learning. With the goal of aiding both the interpretability of dissimilarity measures computed through the Gromov–Wasserstein distance and the assessment of the approximation quality of computational techniques designed to estimate the Gromov–Wasserstein distance, we determine the precise value of a certain variant of the Gromov–Wasserstein distance between unit spheres of different dimensions. Indeed, we consider a two-parameter family \(\{d_{{{\text {GW}}}p,q}\}_{p,q=1}^{\infty }\) of Gromov–Wasserstein distances between metric measure spaces. By exploiting a suitable interaction between specific values of the parameters p and q and the metric of the underlying spaces, we are able to determine the exact value of the distance \(d_{{{\text {GW}}}4,2}\) between all pairs of unit spheres of different dimensions endowed with their Euclidean distance and their uniform measure.

Gromov-Wasserstein 距离（通常 Wasserstein 距离的推广）允许比较在可能不同的度量空间上定义的概率度量。最近，这种距离概念在数据科学和机器学习中得到了多种应用。为了帮助通过 Gromov-Wasserstein 距离计算的相异性度量的可解释性以及评估旨在估计 Gromov-Wasserstein 距离的计算技术的近似质量，我们确定了 Gromov-Wasserstein 某个变体的精确值。不同维度的单位球体之间的 Wasserstein 距离。事实上，我们考虑 Gromov-Wasserstein 距离的二参数族\(\{d_{{{\text {GW}}}p,q}\}_{p,q=1}^{\infty }\)公制测量空间之间。通过利用参数p和q的特定值与底层空间的度量之间的适当交互，我们能够确定距离\(d_{{{\text {GW}}}4,2}的精确值\)在具有欧氏距离和统一测度的所有不同维度的单位球体对之间。