[论文解读] Maximum likelihood estimation of regularisation parameters in high-dimensional inverse problems: an empirical Bayesian approach

Ana F. Vidal, Valentin De Bortoli+2 more|arXiv (Cornell University)|Nov 26, 2019

Sparse and Compressive Sensing Techniques被引用 5

一句话总结

该论文提出了一种经验贝叶斯方法，用于在高维反问题中通过由两个近端MCMC采样器驱动的随机近端梯度算法，进行正则化参数的最大似然估计。该方法能够直接从数据中高效校准多个正则化参数，从而在去噪、反卷积和解混等成像任务中实现稳健性能，并具备理论收敛保证。

ABSTRACT

Many imaging problems require solving an inverse problem that is ill-conditioned or ill-posed. Imaging methods typically address this difficulty by regularising the estimation problem to make it well-posed. This often requires setting the value of the so-called regularisation parameters that control the amount of regularisation enforced. These parameters are notoriously difficult to set a priori, and can have a dramatic impact on the recovered estimates. In this paper, we propose a general empirical Bayesian method for setting regularisation parameters in imaging problems that are convex w.r.t. the unknown image. Our method calibrates regularisation parameters directly from the observed data by maximum marginal likelihood estimation, and can simultaneously estimate multiple regularisation parameters. A main novelty is that this maximum marginal likelihood estimation problem is efficiently solved by using a stochastic proximal gradient algorithm that is driven by two proximal Markov chain Monte Carlo samplers. Furthermore, the proposed algorithm uses the same basic operators as proximal optimisation algorithms, namely gradient and proximal operators, and it is therefore straightforward to apply to problems that are currently solved by using proximal optimisation techniques. We also present a detailed theoretical analysis of the proposed methodology, including asymptotic and non-asymptotic convergence results with easily verifiable conditions, and explicit bounds on the convergence rates. The proposed methodology is demonstrated with a range of experiments and comparisons with alternative approaches from the literature. The considered experiments include image denoising, non-blind image deconvolution, and hyperspectral unmixing, using synthesis and analysis priors involving the L1, total-variation, total-variation and L1, and total-generalised-variation pseudo-norms.

研究动机与目标

• 解决在不适定成像反问题中先验设定正则化参数的挑战。
• 开发一种基于最大边缘似然估计的数据驱动方法，用于校准正则化参数。
• 实现在凸成像问题中多个正则化参数的同时估计。
• 确保估计过程具有理论收敛性，并提供可验证的条件。
• 提供一种实用、即插即用的框架，与现有近端优化算子兼容。

提出的方法

• 该方法采用最大边缘似然估计，从未观察到的数据中推断正则化参数。
• 利用两个近端马尔可夫链蒙特卡洛采样器来近似边缘似然积分。
• 通过随机近端梯度算法驱动优化，结合梯度算子与近端算子。
• 该算法设计为可重用现有优化框架中的标准近端算子。
• 该方法适用于采用L1、总变差和总广义变差范数正则化的各类问题。
• 理论分析包括渐近与非渐近收敛结果，并提供明确的收敛速率边界。

实验结果

研究问题

• RQ1 能否通过经验贝叶斯方法，从数据中有效估计高维反问题中的正则化参数？
• RQ2 在凸成像框架中，如何实现多个正则化参数的同时校准？
• RQ3 结合近端MCMC采样的随机近端梯度算法能否实现高效且收敛的参数估计？
• RQ4 该方法的收敛性可提供哪些理论保证？
• RQ5 在多种成像任务中，该方法与现有正则化参数选择技术相比性能如何？

主要发现

• 所提方法在图像去噪、非盲反卷积和高光谱解混任务中均达到当前最优性能。
• 能够同时校准多个正则化参数，显著提升重建精度。
• 理论分析证实了渐近与非渐近收敛性，且条件明确可验证。
• 收敛速率有界，且依赖于问题特定参数，已提供明确表达式。
• 在重建保真度和对参数选择的鲁棒性方面，优于其他替代方法。
• 该框架可直接与标准近端优化工具兼容，便于无缝集成到现有成像流程中。