# How Regularization Terms Make Invertible Neural Networks Bayesian Point Estimators

arXiv: https://arxiv.org/abs/2510.26704 · HTML: https://mlsanity.com/paper/2510.26704/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 31 references that are in the map.
- First posted on arXiv 30 Oct 2025.

## Builds on (most cited first, 18 of 18)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration](https://mlsanity.com/paper/2303.11435/) (2023)
- [Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problems](https://mlsanity.com/paper/1704.03488/) (2017)
- [Improving DNN Robustness to Adversarial Attacks using Jacobian Regularization](https://mlsanity.com/paper/1803.08680/) (2018)
- [Semi-Supervised Learning with Normalizing Flows](https://mlsanity.com/paper/1912.13025/) (2019)
- [Understanding and Mitigating Exploding Inverses in Invertible Neural Networks](https://mlsanity.com/paper/2006.09347/) (2020)
- [Composing Normalizing Flows for Inverse Problems](https://mlsanity.com/paper/2002.11743/) (2020)
- [Solving inverse problems using conditional invertible neural networks](https://mlsanity.com/paper/2007.15849/) (2020)
- [Semi-supervised Invertible Neural Operators for Bayesian Inverse Problems](https://mlsanity.com/paper/2209.02772/) (2022)
- [Conditional Invertible Neural Networks for Medical Imaging](https://mlsanity.com/paper/2110.14520/) (2021)
- [Invertible ResNets for Inverse Imaging Problems: Competitive Performance with Provable Regularization Properties](https://mlsanity.com/paper/2409.13482/) (2024)

## Abstract (arXiv)

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.