In this work, we construct optimal low-rank approximations for the Gaussian posterior distribution in linear Gaussian inverse problems. The parameter space is a separable Hilbert space of possibly infinite dimension, and the data space is assumed to be finite-dimensional. We consider various types of approximation families for the posterior. We first consider approximate posteriors in which the means vary among a class of either structure-preserving or structure-ignoring low-rank transformations of the data, and in which the posterior covariance is kept fixed. We give necessary and sufficient conditions for these approximating posteriors to be equivalent to the exact posterior, for all possible realisations of the data simultaneously. For such approximations, we measure approximation error with the Kullback-Leibler, R\'enyi and Amari $\alpha$-divergences for $\alpha\in(0,1)$, and with the Hellinger distance, all averaged over the data distribution. With these losses, we find the optimal approximations and formulate an equivalent condition for their uniqueness, extending the work in finite dimensions of Spantini et al. (SIAM J. Sci. Comput. 2015). We then consider joint approximation of the mean and covariance, by also varying the posterior covariance over the low-rank updates considered in Part I of this work. For the reverse Kullback-Leibler divergence, we show that the separate optimal approximations of the mean and of the covariance can be combined to yield an optimal joint approximation of the mean and covariance. In addition, we interpret the joint approximation with the optimal structure-ignoring approximate mean in terms of an optimal projector in parameter space.

翻译：在本工作中，我们针对线性高斯反问题中的高斯后验分布构建了最优低秩逼近。参数空间为可能无限维的可分希尔伯特空间，数据空间假定为有限维。我们考虑了多种类型的后验逼近族。首先考虑近似后验分布，其中均值在保持结构或忽略结构的低秩数据变换类中变化，而后验协方差保持固定。我们给出了这些近似后验对所有可能数据实现同时等价于精确后验的充分必要条件。对于此类逼近，我们采用Kullback-Leibler散度、Rényi散度、Amari $\alpha$-散度（$\alpha\in(0,1)$）以及Hellinger距离作为误差度量，并取其在数据分布上的平均值。基于这些损失函数，我们找到了最优逼近并建立了其唯一性的等价条件，推广了Spantini等人（SIAM J. Sci. Comput. 2015）在有限维情形下的工作。随后我们考虑均值与协方差的联合逼近，通过将后验协方差在本文第一部分所考虑的低秩更新范围内变化。对于反向Kullback-Leibler散度，我们证明均值与协方差的独立最优逼近可以组合得到均值与协方差的最优联合逼近。此外，我们通过参数空间中的最优投影算子，对具有最优忽略结构近似均值的联合逼近进行了解释。