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
Connecting Hamilton--Jacobi partial differential equations with maximum a posteriori and posterior mean estimators for some non-convex priors
摘要：
许多成像问题可以用逆问题的形式表示，逆问题可以表示为有限维最优化问题。这些最优化问题通常包括最小化数据保真度和正则化项之和。在[23，26]中，这些最优化问题和（多时间）Hamilton-Jacobi偏微分方程之间的联系已经在数据保真度和正则化项的凸性假设下提出。特别地，在这些凸性假设下，可以得到极小元的一些表示公式。从贝叶斯的角度来看，这样的最小值可以被视为最大后验估计。在这一章中，我们考虑了一类非凸正则化，并证明了类似的表示公式的极小也可以得到。这是通过利用最小-加代数技术来实现的，这些技术最初是为求解在最优控制中产生的某些Hamilton-Jacobi偏微分方程而开发的。请注意，粘性Hamilton-Jacobi偏微分方程与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之间的联系已在[25]中突出显示。我们还提出了类似的结果，某些贝叶斯后验均值估计与高斯数据保真度和某些非对数凹先验使用模拟最小代数技术。
英文摘要：
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 [23,26], 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 [25]. 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.
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