This article explores the optimization of variational approximations for posterior covariances of Gaussian multiway arrays. To achieve this, we establish a natural differential geometric optimization framework on the space using the pullback of the affine-invariant metric. In the case of a truly separable covariance, we demonstrate a joint approximation in the multiway space outperforms a mean-field approximation in optimization efficiency and provides a superior approximation to an unstructured Inverse-Wishart posterior under the average Mahalanobis distance of the data while maintaining a multiway interpretation. We moreover establish efficient expressions for the Euclidean and Riemannian gradients in both cases of the joint and mean-field approximation. We end with an analysis of commodity trade data.

翻译：本文探讨了高斯多路阵列后验协方差的变分近似优化问题。为此，我们在该空间上利用仿射不变度量的拉回建立了自然的微分几何优化框架。在真实可分离协方差的情形下，我们证明了多路空间中的联合近似在优化效率上优于平均场近似，并在保持多路解释的同时，通过数据的平均马氏距离为无结构逆威沙特后验提供了更优的近似。此外，我们针对联合近似与平均场近似两种情形，分别建立了欧几里得梯度与黎曼梯度的有效表达式。最后，我们对商品贸易数据进行了分析。