# Variational Mixture of Normalizing Flows

arXiv: https://arxiv.org/abs/2009.00585 · HTML: https://mlsanity.com/paper/2009.00585/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Sep 2020.

## Built on by (most cited first, 15 of 15)

- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [Go with the Flows: Mixtures of Normalizing Flows for Point Cloud Generation and Reconstruction](https://mlsanity.com/paper/2106.03135/) (2021)
- [On the expressivity of bi-Lipschitz normalizing flows](https://mlsanity.com/paper/2107.07232/) (2021)
- [Normalizing flow sampling with Langevin dynamics in the latent space](https://mlsanity.com/paper/2305.12149/) (2023)
- [Cooperation in the Latent Space: The Benefits of Adding Mixture Components in Variational Autoencoders](https://mlsanity.com/paper/2209.15514/) (2022)
- [CEIP: Combining Explicit and Implicit Priors for Reinforcement Learning with Demonstrations](https://mlsanity.com/paper/2210.09496/) (2022)
- [SESaMo: Symmetry-Enforcing Stochastic Modulation for Normalizing Flows](https://mlsanity.com/paper/2505.19619/) (2025)
- [Fitting large mixture models using stochastic component selection](https://mlsanity.com/paper/2110.04776/) (2021)
- [Marginalization Consistent Probabilistic Forecasting of Irregular Time Series via Mixture of Separable flows](https://mlsanity.com/paper/2406.07246/) (2024)
- [Time Series Clustering with General State Space Models via Stochastic Variational Inference](https://mlsanity.com/paper/2407.00429/) (2024)
- [Scalable Expectation Estimation with Subtractive Mixture Models](https://mlsanity.com/paper/2503.21346/) (2025)
- [Adaptive Heterogeneous Mixtures of Normalising Flows for Robust Variational Inference](https://mlsanity.com/paper/2510.02056/) (2025)
- [Rethinking the Flow-Based Gradual Domain Adaptation: A Semi-Dual Optimal Transport Perspective](https://mlsanity.com/paper/2602.01179/) (2026)
- [Phase-Type Variational Autoencoders for Heavy-Tailed Data](https://mlsanity.com/paper/2603.01800/) (2026)
- [Stable Global Weighting of Flow Mixtures using Simplex Exponential Moving Average](https://mlsanity.com/paper/2607.03809/) (2026)

## Abstract (arXiv)

In the past few years, deep generative models, such as generative adversarial networks autociteGAN, variational autoencoders autocitevaepaper, and their variants, have seen wide adoption for the task of modelling complex data distributions. In spite of the outstanding sample quality achieved by those early methods, they model the target distributions implicitly, in the sense that the probability density functions induced by them are not explicitly accessible. This fact renders those methods unfit for tasks that require, for example, scoring new instances of data with the learned distributions. Normalizing flows have overcome this limitation by leveraging the change-of-variables formula for probability density functions, and by using transformations designed to have tractable and cheaply computable Jacobians. Although flexible, this framework lacked (until recently autocitessemisuplearningnflows, RAD) a way to introduce discrete structure (such as the one found in mixtures) in the models it allows to construct, in an unsupervised scenario. The present work overcomes this by using normalizing flows as components in a mixture model and devising an end-to-end training procedure for such a model. This procedure is based on variational inference, and uses a variational posterior parameterized by a neural network. As will become clear, this model naturally lends itself to (multimodal) density estimation, semi-supervised learning, and clustering. The proposed model is illustrated on two synthetic datasets, as well as on a real-world dataset. Keywords: Deep generative models, normalizing flows, variational inference, probabilistic modelling, mixture models.