Optimal low-rank approximations for linear Gaussian inverse problems on Hilbert spaces, Part I: posterior covariance approximation
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arXiv
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| Format: | Preprint |
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2025
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| author | Carere, Giuseppe Lie, Han Cheng |
| author_facet | Carere, Giuseppe Lie, Han Cheng |
| contents | 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 Rényi divergences, the Amari $α$-divergences for $α\in(0,1)$, 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 discretization independent results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_24020 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal low-rank approximations for linear Gaussian inverse problems on Hilbert spaces, Part I: posterior covariance approximation Carere, Giuseppe Lie, Han Cheng Statistics Theory Probability 28C20, 47A58, 60G15, 62F15, 62G05 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 Rényi divergences, the Amari $α$-divergences for $α\in(0,1)$, 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 discretization independent results. |
| title | Optimal low-rank approximations for linear Gaussian inverse problems on Hilbert spaces, Part I: posterior covariance approximation |
| topic | Statistics Theory Probability 28C20, 47A58, 60G15, 62F15, 62G05 |
| url | https://arxiv.org/abs/2503.24020 |