# Stabilizing Invertible Neural Networks Using Mixture Models

arXiv: https://arxiv.org/abs/2009.02994 · HTML: https://mlsanity.com/paper/2009.02994/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Sep 2020.

## Built on by (most cited first, 30 of 30)

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- [Improving Lipschitz-Constrained Neural Networks by Learning Activation Functions](https://mlsanity.com/paper/2210.16222/) (2022)
- [Generalized Normalizing Flows via Markov Chains](https://mlsanity.com/paper/2111.12506/) (2021)
- [PatchNR: Learning from Very Few Images by Patch Normalizing Flow Regularization](https://mlsanity.com/paper/2205.12021/) (2022)
- [Approximation of Lipschitz Functions using Deep Spline Neural Networks](https://mlsanity.com/paper/2204.06233/) (2022)
- [Conditional Generative Models are Provably Robust: Pointwise Guarantees for Bayesian Inverse Problems](https://mlsanity.com/paper/2303.15845/) (2023)
- [A Review of Change of Variable Formulas for Generative Modeling](https://mlsanity.com/paper/2308.02652/) (2023)
- [Invertible Neural Networks versus MCMC for Posterior Reconstruction in Grazing Incidence X-Ray Fluorescence](https://mlsanity.com/paper/2102.03189/) (2021)
- [Learning from small data sets: Patch-based regularizers in inverse problems for image reconstruction](https://mlsanity.com/paper/2312.16611/) (2023)
- [Stability of Image-Reconstruction Algorithms](https://mlsanity.com/paper/2206.07128/) (2022)
- [Normalizing flow sampling with Langevin dynamics in the latent space](https://mlsanity.com/paper/2305.12149/) (2023)
- [Topology-Matching Normalizing Flows for Out-of-Distribution Detection in Robot Learning](https://mlsanity.com/paper/2311.06481/) (2023)
- [Slicing the Gaussian Mixture Wasserstein Distance](https://mlsanity.com/paper/2504.08544/) (2025)
- [Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces](https://mlsanity.com/paper/2205.14627/) (2022)
- [Stable Training of Normalizing Flows for High-dimensional Variational Inference](https://mlsanity.com/paper/2402.16408/) (2024)
- [Proximal Residual Flows for Bayesian Inverse Problems](https://mlsanity.com/paper/2211.17158/) (2022)
- [(Almost) Smooth Sailing: Towards Numerical Stability of Neural Networks Through Differentiable Regularization of the Condition Number](https://mlsanity.com/paper/2410.00169/) (2024)
- [NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problems](https://mlsanity.com/paper/2304.08342/) (2023)
- [Convergence and Recovery Guarantees of Unsupervised Neural Networks for Inverse Problems](https://mlsanity.com/paper/2309.12128/) (2023)
- [Generalized Wasserstein Flow Matching: Transport Plans, Everywhere, All at Once](https://mlsanity.com/paper/2605.08424/) (2026)
- [On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networks](https://mlsanity.com/paper/2308.09367/) (2023)
- [Piecewise Normalizing Flows](https://mlsanity.com/paper/2305.02930/) (2023)
- [Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overview](https://mlsanity.com/paper/2402.12072/) (2024)
- [Noise-adapted Neural Operator for Robust Non-Line-of-Sight Imaging](https://mlsanity.com/paper/2508.09655/) (2025)
- [Adapting Noise to Data: Generative Flows from 1D Processes](https://mlsanity.com/paper/2510.12636/) (2025)
- [Simplifying Flow Matching Transformations with Low-Rank Mixture Models](https://mlsanity.com/paper/2606.29724/) (2026)

## Abstract (arXiv)

In this paper, we analyze the properties of invertible neural networks, which provide a way of solving inverse problems. Our main focus lies on investigating and controlling the Lipschitz constants of the corresponding inverse networks. Without such an control, numerical simulations are prone to errors and not much is gained against traditional approaches. Fortunately, our analysis indicates that changing the latent distribution from a standard normal one to a Gaussian mixture model resolves the issue of exploding Lipschitz constants. Indeed, numerical simulations confirm that this modification leads to significantly improved sampling quality in multimodal applications.