We study the entropic Gromov-Wasserstein and its unbalanced version between (unbalanced) Gaussian distributions with different dimensions. When the metric is the inner product, which we refer to as inner product Gromov-Wasserstein (IGW), we demonstrate that the optimal transportation plans of entropic IGW and its unbalanced variant are (unbalanced) Gaussian distributions. Via an application of von Neumann's trace inequality, we obtain closed-form expressions for the entropic IGW between these Gaussian distributions. Finally, we consider an entropic inner product Gromov-Wasserstein barycenter of multiple Gaussian distributions. We prove that the barycenter is a Gaussian distribution when the entropic regularization parameter is small. We further derive a closed-form expression for the covariance matrix of the barycenter.

翻译：我们研究了Gromov-Wasserstein 及其不同维度的(不平衡的)高斯分布之间的偏移版本。当该指标是内产物时,我们称之为内产物Gromov-Wasserstein(IGW),我们证明,英制IGW及其不平衡变体的最佳运输计划是(不平衡的)高斯恩分布。通过 von Neumann 的微量不平等的应用,我们获得了这些高斯的分布之间的超形式表达式。最后,我们考虑了多种高斯分布的内产物Gromov-Wasserstein barycenter。我们证明,当伦式正规化参数小时,该中转器是一种高斯的分布式。我们进一步得出了这些高斯中心常态矩阵的封闭式表达式表达式。