正则化项让可逆神经网络实现贝叶斯点估计，稳定可解释。
How Regularization Terms Make Invertible Neural Networks Bayesian Point Estimators
- 引入两种正则化项，使网络反演后对应贝叶斯均值与最大后验估计。
- 理论分析揭示损失函数如何塑造前向算子和反演重建映射。
- 数值实验验证方法在数据依赖性上稳定且可解释，适合逆问题建模。
可逆神经网络因其内在稳定性与可解释性，在逆问题中备受关注。近期研究从贝叶斯视角分析了近似重建映射或前向算子的优化策略，但各有局限。本文提出并分析两种训练正则化项：前者反演后对应后验均值，后者类似最大后验（MAP）估计。理论分析表明，每种损失函数均能塑造学习到的前向算子及其逆映射的性质。数值实验验证了这些正则化项能以稳定且可解释的方式引入数据依赖性，支持其作为贝叶斯点估计器的有效性。
原文摘要 · Abstract (English)
Can regularization terms in the training of invertible neural networks lead to known Bayesian point estimators in reconstruction? Invertible networks are attractive for inverse problems due to their inherent stability and interpretability. Recently, optimization strategies for invertible neural networks that approximate either a reconstruction map or the forward operator have been studied from a Bayesian perspective, but each has limitations. To address this, we introduce and analyze two regularization terms for the network training that, upon inversion of the network, recover properties of classical Bayesian point estimators: while the first can be connected to the posterior mean, the second resembles the MAP estimator. Our theoretical analysis characterizes how each loss shapes both the learned forward operator and its inverse reconstruction map. Numerical experiments support our findings and demonstrate how these loss-term regularizers introduce data-dependence in a stable and interpretable way.
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