Connecting Hamilton-Jacobi Partial Differential Equations with Maximum a Posteriori and Posterior Mean Estimators for Some Non-convex Priors.
Connecting Hamilton-Jacobi Partial Differential Equations with Maximum a Posteriori and Posterior Mean Estimators for Some Non-convex Priors.
复制标题
将 Hamilton-Jacobi 偏微分方程与某些非凸先验的最大后验和后验均值估计器连接起来。
DOI：
10.1007/978-3-030-03009-4_56-1
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Meng, T.
中科院分区：
文献类型：
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作者：
Darbon, J.;Langlois, G.P.;Meng, T.
许多成像问题可以用有限维优化问题表示的反问题来表示。这些优化问题通常包括最小化数据保真度和正则化项的总和。在 Darbon (SIAM J. Imag. Sci. 8:2268–2293, 2015)、Darbon 和 Meng 中，(关于成像科学中的分解模型和多次 Hamilton-Jacobi 偏微分方程，arXiv 预印本 arXiv:1906.09502, 2019)，已经提出了这些优化问题与（多次）Hamilton-Jacobi 偏微分方程之间的联系 在数据保真度和正则化项的凸性假设下。特别地，在这些凸性假设下，可以获得最小化器的一些表示公式。从贝叶斯的角度来看，这样的最小化器可以被视为最大后验估计器。在本章中，我们考虑一类非凸正则化，并表明也可以获得类似的最小化器表示公式。这是通过利用最小加代数技术来实现的，这些技术最初是为了求解最优控制中出现的某些汉密尔顿-雅可比偏微分方程而开发的。请注意，Darbon 和 Langlois 中强调了粘性 Hamilton-Jacobi 偏微分方程和具有高斯数据保真度项和对数凹先验的贝叶斯后验均值估计量之间的联系（关于成像科学中的贝叶斯后验均值估计量和 Hamilton-Jacobi 偏微分方程，arXiv 预印本 arXiv:2003.05572， 2020）。我们还使用类似的最小加代数技术，对具有高斯数据保真度的某些贝叶斯后验均值估计器和某些非对数凹先验提出了类似的结果。
Many imaging problems can be formulated as inverse problems expressed as finite-dimensional optimization problems. These optimization problems generally consist of minimizing the sum of a data fidelity and regularization terms. In Darbon (SIAM J. Imag. Sci. 8:2268–2293, 2015), Darbon and Meng, (On decomposition models in imaging sciences and multi-time Hamilton-Jacobi partial differential equations, arXiv preprint arXiv:1906.09502, 2019), connections between these optimization problems and (multi-time) Hamilton-Jacobi partial differential equations have been proposed under the convexity assumptions of both the data fidelity and regularization terms. In particular, under these convexity assumptions, some representation formulas for a minimizer can be obtained. From a Bayesian perspective, such a minimizer can be seen as a maximum a posteriori estimator. In this chapter, we consider a certain class of non-convex regularizations and show that similar representation formulas for the minimizer can also be obtained. This is achieved by leveraging min-plus algebra techniques that have been originally developed for solving certain Hamilton-Jacobi partial differential equations arising in optimal control. Note that connections between viscous Hamilton-Jacobi partial differential equations and Bayesian posterior mean estimators with Gaussian data fidelity terms and log-concave priors have been highlighted in Darbon and Langlois, (On Bayesian posterior mean estimators in imaging sciences and Hamilton-Jacobi partial differential equations, arXiv preprint arXiv:2003.05572, 2020). We also present similar results for certain Bayesian posterior mean estimators with Gaussian data fidelity and certain non-log-concave priors using an analogue of min-plus algebra techniques.