# Scalable Normalizing Flows for Permutation Invariant Densities

arXiv: https://arxiv.org/abs/2010.03242 · HTML: https://mlsanity.com/paper/2010.03242/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Oct 2020.

## Built on by (most cited first, 22 of 22)

- [Neural Flows: Efficient Alternative to Neural ODEs](https://mlsanity.com/paper/2110.13040/) (2021)
- [Structure-preserving GANs](https://mlsanity.com/paper/2202.01129/) (2022)
- [Sample Complexity of Probability Divergences under Group Symmetry](https://mlsanity.com/paper/2302.01915/) (2023)
- [Sample Complexity Bounds for Estimating Probability Divergences under Invariances](https://mlsanity.com/paper/2311.02868/) (2023)
- [Equivariant Finite Normalizing Flows](https://mlsanity.com/paper/2110.08649/) (2021)
- [Semi-Discrete Normalizing Flows through Differentiable Tessellation](https://mlsanity.com/paper/2203.06832/) (2022)
- [On permutation-invariant neural networks](https://mlsanity.com/paper/2403.17410/) (2024)
- [Deep Variational Free Energy Approach to Dense Hydrogen](https://mlsanity.com/paper/2209.06095/) (2022)
- [Conditional Permutation Invariant Flows](https://mlsanity.com/paper/2206.09021/) (2022)
- [Equivariant Neural Simulators for Stochastic Spatiotemporal Dynamics](https://mlsanity.com/paper/2305.14286/) (2023)
- [Learning with Exact Invariances in Polynomial Time](https://mlsanity.com/paper/2502.19758/) (2025)
- [Audio synthesizer inversion in symmetric parameter spaces with approximately equivariant flow matching](https://mlsanity.com/paper/2506.07199/) (2025)
- [Boltzmann generators for amorphous particle systems](https://mlsanity.com/paper/2512.16607/) (2025)
- [Statistical Guarantees of Group-Invariant GANs](https://mlsanity.com/paper/2305.13517/) (2023)
- [Efficient Learning and Symmetry Discovery under Exact Invariances](https://mlsanity.com/paper/2609.07031/) (2026)
- [Towards Cost Sensitive Decision Making](https://mlsanity.com/paper/2410.03892/) (2024)
- [Generative Unordered Flow for Set-Structured Data Generation](https://mlsanity.com/paper/2501.17770/) (2025)
- [Probabilistic Forecasting of Irregular Time Series via Conditional Flows](https://mlsanity.com/paper/2402.06293/) (2024)
- [Differentiable free energy surface: a variational approach to directly observing rare events using generative deep-learning models](https://mlsanity.com/paper/2604.09769/) (2026)
- [Marginalization Consistent Probabilistic Forecasting of Irregular Time Series via Mixture of Separable flows](https://mlsanity.com/paper/2406.07246/) (2024)
- [StAD: Stein Amortized Divergence for Fast Likelihoods with Diffusion and Flow](https://mlsanity.com/paper/2605.16486/) (2026)
- [Grand Canonical Generators](https://mlsanity.com/paper/2610.00683/) (2026)

## Abstract (arXiv)

Modeling sets is an important problem in machine learning since this type of data can be found in many domains. A promising approach defines a family of permutation invariant densities with continuous normalizing flows. This allows us to maximize the likelihood directly and sample new realizations with ease. In this work, we demonstrate how calculating the trace, a crucial step in this method, raises issues that occur both during training and inference, limiting its practicality. We propose an alternative way of defining permutation equivariant transformations that give closed form trace. This leads not only to improvements while training, but also to better final performance. We demonstrate the benefits of our approach on point processes and general set modeling.