A Variational Inference Approach to Inverse Problems with Gamma Hyperpriors

Shiv Agrawal; Hwanwoo Kim; Danie Sanz-AlonsoAlexander Strang

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A Variational Inference Approach to Inverse Problems with Gamma Hyperpriors

机译：Gamma超先验逆问题的变分推理方法

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Hierarchical models with gamma hyperpriors provide a flexible, sparsity-promoting framework to bridge L~1 and L~2 regularizations in Bayesian formulations to inverse problems. Despite the Bayesian motivation for these models, existing methodologies are limited to maximum a posteriori estimation. The potential to perform uncertainty quantification has not yet been realized. This paper introduces a variational iterative alternating scheme for hierarchical inverse problems with gamma hyperpriors. The proposed variational inference approach yields accurate reconstruction, provides meaningful uncertainty quantification, and is easy to implement. In addition, it lends itself naturally to conduct model selection for the choice of hyperparameters. We illustrate the performance of our methodology in several computed examples, including a deconvolution problem and sparse identification of dynamical systems from time series data.

机译：层次模型与伽马hyperpriors提供一种灵活的、sparsity-promoting框架桥L ~ 1和L ~ 2在贝叶斯合法化配方的逆问题。这些模型的贝叶斯动机,现有的方法仅限于最大后验估计。量化尚未实现。介绍了一种变分迭代交替方案分层逆γhyperpriors的问题。变分推理方法准确重建,提供有意义的不确定性量化,容易实现。另外,它本身自然地进行模型选择的选择hyperparameters。我们的方法在几个计算实例,包括一个反褶积问题和稀疏动力系统的识别系列数据。

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《SIAM/ASA Journal on Uncertainty Quantification》 |2022年第4期|1533-1559|共27页
作者
Shiv Agrawal; Hwanwoo Kim; Danie Sanz-AlonsoAlexander Strang;
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Department of Statistics, University of Chicago, Chicago, IL 60637 USA;

Committee on Computational and Applied Mathematics, University of Chicago, Chicago, IL 60637 USA;

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中图分类计量学;
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A Variational Inference Approach to Inverse Problems with Gamma Hyperpriors

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