For linear inverse problems with Gaussian priors and Gaussian observation noise, the posterior is Gaussian, with mean and covariance determined by the conditioning formula. Using the Feldman-Hajek theorem, we analyse the prior-to-posterior update and its low-rank approximation for infinite-dimensional Hilbert parameter spaces and finite-dimensional observations. We show that the posterior distribution differs from the prior on a finite-dimensional subspace, and construct low-rank approximations to the posterior covariance, while keeping the mean fixed. Since in infinite dimensions, not all low-rank covariance approximations yield approximate posterior distributions which are equivalent to the posterior and prior distribution, we characterise the low-rank covariance approximations which do yield this equivalence, and their respective inverses, or `precisions'. For such approximations, a family of measure approximation problems is solved by identifying the low-rank approximations which are optimal for various losses simultaneously. These loss functions include the family of R\'enyi divergences, the Amari $\alpha$-divergences for $\alpha\in(0,1)$, the Hellinger metric and the Kullback-Leibler divergence. Our results extend those of Spantini et al. (SIAM J. Sci. Comput. 2015) to Hilbertian parameter spaces, and provide theoretical underpinning for the construction of low-rank approximations of discretised versions of the infinite-dimensional inverse problem, by formulating discretisation independent results.

翻译：对于具有高斯先验与高斯观测噪声的线性反问题，后验分布为高斯分布，其均值与协方差由条件公式确定。利用费尔德曼-哈耶克定理，我们分析了无限维希尔伯特参数空间与有限维观测情形下的先验至后验更新过程及其低秩逼近。我们证明后验分布与先验分布在有限维子空间上存在差异，并构建了后验协方差的低秩逼近，同时保持均值不变。由于在无限维空间中，并非所有低秩协方差逼近都能产生与后验及先验分布等价的后验分布近似，我们刻画了能够实现这种等价性的低秩协方差逼近及其对应的逆矩阵（即“精度矩阵”）。针对此类逼近，通过识别在多种损失函数下同时最优的低秩逼近，我们解决了一类测度逼近问题。这些损失函数包括Rényi散度族、α∈(0,1)时的Amari α-散度、Hellinger距离以及Kullback-Leibler散度。我们的结果将Spantini等人（SIAM J. Sci. Comput. 2015）的研究推广至希尔伯特参数空间，并通过建立离散化无关的结论，为无限维反问题离散化版本的低秩逼近构建提供了理论基础。