A hierarchical Bayesian perspective on majorization-minimization for non-convex sparse regression: application to M/EEG source imaging
A hierarchical Bayesian perspective on majorization-minimization for non-convex sparse regression: application to M/EEG source imaging
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DOI：
10.1088/1361-6420/aac9b3
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发表时间：
2018-08-01
期刊：
影响因子：
2.1
通讯作者：
Gramfort, Alexandre
中科院分区：
文献类型：
--
作者：
Bekhti, Yousra;Lucka, Felix;Gramfort, Alexandre
优化最小化（MM）是一种标准的迭代优化技术，它包括最小化一系列凸代理泛函。MM方法已经特别成功地解决了逆问题和统计机器学习问题，其中正则化项是稀疏促进凹函数。然而，由于非凸性，MM找到的解决方案取决于它的初始化。均匀初始化是最自然和最常用的策略，因为它归结为在第一次MM迭代中同等地惩罚所有系数。然而，这种任意的选择可能会导致不满意的结果，严重欠定逆问题，如源成像与磁和脑电图（M/EEG）。层次贝叶斯模型（HBM）的框架是一种替代方法来编码稀疏性。这项工作表明，对于某些分层模型，一个简单的交替方案来计算完全贝叶斯最大后验概率（MAP）估计导致与标准MM策略完全相同的更新序列（见自适应套索）。有了这个平行概述，我们展示了如何改进这些MM技术，探测多峰后验密度使用马尔可夫链蒙特-卡罗（MCMC）技术。首先，我们表明，这些样本可以提供充分知情的初始化，帮助MM计划，以达到更好的局部极小值。其次，我们展示了它是如何揭示后验分布的不同模式，以探索和量化这种不适定推理过程的内在不确定性和模糊性。在M/EEG的上下文中，每个模式对应于神经源的合理配置，这对于数据解释至关重要，特别是在临床环境中。模拟和真实的数据集上的结果显示了传感器的数量或类型如何影响估计的不确定性。
Majorization-minimization (MM) is a standard iterative optimization technique which consists in minimizing a sequence of convex surrogate functionals. MM approaches have been particularly successful to tackle inverse problems and statistical machine learning problems where the regularization term is a sparsity-promoting concave function. However, due to non-convexity, the solution found by MM depends on its initialization. Uniform initialization is the most natural and often employed strategy as it boils down to penalizing all coefficients equally in the first MM iteration. Yet, this arbitrary choice can lead to unsatisfactory results in severely underdetermined inverse problems such as source imaging with magneto-and electro-encephalography (M/EEG). The framework of hierarchical Bayesian modeling (HBM) is an alternative approach to encode sparsity. This work shows that for certain hierarchical models, a simple alternating scheme to compute fully Bayesian maximum a posteriori (MAP) estimates leads to the exact same sequence of updates as a standard MM strategy (see the adaptive lasso). With this parallel outlined, we show how to improve upon these MM techniques by probing the multimodal posterior density using Markov Chain Monte-Carlo (MCMC) techniques. Firstly, we show that these samples can provide well-informed initializations that help MM schemes to reach better local minima. Secondly, we demonstrate how it can reveal the different modes of the posterior distribution in order to explore and quantify the inherent uncertainty and ambiguity of such ill-posed inference procedure. In the context of M/EEG, each mode corresponds to a plausible configuration of neural sources, which is crucial for data interpretation, especially in clinical contexts. Results on both simulations and real datasets show how the number or the type of sensors affect the uncertainties on the estimates.