# Spatiotemporal Besov Priors for Bayesian Inverse Problems

arXiv: https://arxiv.org/abs/2306.16378 · HTML: https://mlsanity.com/paper/2306.16378/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 122 references that are in the map.
- First posted on arXiv 28 Jun 2023.

## Builds on (most cited first, 18 of 18)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Hyperband: A Novel Bandit-Based Approach to Hyperparameter Optimization](https://mlsanity.com/paper/1603.06560/) (2016)
- [BOHB: Robust and Efficient Hyperparameter Optimization at Scale](https://mlsanity.com/paper/1807.01774/) (2018)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [A Haar Wavelet-Based Perceptual Similarity Index for Image Quality Assessment](https://mlsanity.com/paper/1607.06140/) (2016)
- [Sparsity information and regularization in the horseshoe and other shrinkage priors](https://mlsanity.com/paper/1707.01694/) (2017)
- [Dimension-Robust MCMC in Bayesian Inverse Problems](https://mlsanity.com/paper/1803.03344/) (2018)
- [Generalized probabilistic principal component analysis of correlated data](https://mlsanity.com/paper/1808.10868/) (2018)
- [Gaussian orthogonal latent factor processes for large incomplete matrices of correlated data](https://mlsanity.com/paper/2011.10863/) (2020)
- [A Stable Manifold MCMC Method for High Dimensions](https://mlsanity.com/paper/1403.7711/) (2014)
- [A Bayesian Level Set Method for Geometric Inverse Problems](https://mlsanity.com/paper/1504.00313/) (2015)
- [Learning Temporal Evolution of Spatial Dependence with Generalized Spatiotemporal Gaussian Process Models](https://mlsanity.com/paper/1901.04030/) (2019)
- [Bayesian Learning via Q-Exponential Process](https://mlsanity.com/paper/2210.07987/) (2022)
- [A TV-Gaussian prior for infinite-dimensional Bayesian inverse problems and its numerical implementations](https://mlsanity.com/paper/1510.05239/) (2015)
- [Cauchy Markov Random Field Priors for Bayesian Inversion](https://mlsanity.com/paper/2105.12488/) (2021)
- [STEMPO -- dynamic X-ray tomography phantom](https://mlsanity.com/paper/2209.12471/) (2022)
- [Multivariate type G Mat'ern stochastic partial differential equation random fields](https://mlsanity.com/paper/1606.08298/) (2016)

## Built on by (most cited first, 3 of 3)

- [Single-shot Tomography of Discrete Dynamic Objects](https://mlsanity.com/paper/2311.05269/) (2023)
- [SAMCIRT: A Simultaneous Reconstruction and Affine Motion Compensation Technique for Four Dimensional Computed Tomography (4DCT)](https://mlsanity.com/paper/2402.04480/) (2024)
- [Deep Q-Exponential Processes](https://mlsanity.com/paper/2410.22119/) (2024)

## Abstract (arXiv)

Fast development in science and technology has driven the need for proper statistical tools to capture special data features such as abrupt changes or sharp contrast. Many inverse problems in data science require spatiotemporal solutions derived from a sequence of time-dependent objects with these spatial features, e.g., the dynamic reconstruction of computerized tomography (CT) images with edges. Conventional methods based on Gaussian processes (GP) often fall short in providing satisfactory solutions since they tend to offer oversmooth priors. Recently, the Besov process (BP), defined by wavelet expansions with random coefficients, has emerged as a more suitable prior for Bayesian inverse problems of this nature. While BP excels in handling spatial inhomogeneity, it does not automatically incorporate temporal correlation inherited in the dynamically changing objects. In this paper, we generalize BP to a novel spatiotemporal Besov process (STBP) by replacing the random coefficients in the series expansion with stochastic time functions as Q-exponential process (Q-EP) which governs the temporal correlation structure. We thoroughly investigate the mathematical and statistical properties of STBP. Simulations, two limited-angle CT reconstruction examples, a highly non-linear inverse problem involving Navier-Stokes equation, and a spatiotemporal temperature imputation problem are used to demonstrate the advantage of the proposed STBP compared with the classic STGP and a time-uncorrelated approach.