# Woodbury Transformations for Deep Generative Flows

arXiv: https://arxiv.org/abs/2002.12229 · HTML: https://mlsanity.com/paper/2002.12229/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Feb 2020.

## Built on by (most cited first, 10 of 10)

- [Maximum Entropy Reinforcement Learning via Energy-Based Normalizing Flow](https://mlsanity.com/paper/2405.13629/) (2024)
- [ButterflyFlow: Building Invertible Layers with Butterfly Matrices](https://mlsanity.com/paper/2209.13774/) (2022)
- [Robust normalizing flows using Bernstein-type polynomials](https://mlsanity.com/paper/2102.03509/) (2021)
- [Invertible Monotone Operators for Normalizing Flows](https://mlsanity.com/paper/2210.08176/) (2022)
- [Training Energy-Based Normalizing Flow with Score-Matching Objectives](https://mlsanity.com/paper/2305.15267/) (2023)
- [Multi-Resolution Continuous Normalizing Flows](https://mlsanity.com/paper/2106.08462/) (2021)
- [FInC Flow: Fast and Invertible k × k Convolutions for Normalizing Flows](https://mlsanity.com/paper/2301.09266/) (2023)
- [CDFlow: Building Invertible Layers with Circulant and Diagonal Matrices](https://mlsanity.com/paper/2510.25323/) (2025)
- [MixerFlow: MLP-Mixer meets Normalising Flows](https://mlsanity.com/paper/2310.16777/) (2023)
- [Quasi-Linear ICA for Motor Unit Decomposition during Dynamic Contractions](https://mlsanity.com/paper/2406.19581/) (2024)

## Abstract (arXiv)

Normalizing flows are deep generative models that allow efficient likelihood calculation and sampling. The core requirement for this advantage is that they are constructed using functions that can be efficiently inverted and for which the determinant of the function's Jacobian can be efficiently computed. Researchers have introduced various such flow operations, but few of these allow rich interactions among variables without incurring significant computational costs. In this paper, we introduce Woodbury transformations, which achieve efficient invertibility via the Woodbury matrix identity and efficient determinant calculation via Sylvester's determinant identity. In contrast with other operations used in state-of-the-art normalizing flows, Woodbury transformations enable (1) high-dimensional interactions, (2) efficient sampling, and (3) efficient likelihood evaluation. Other similar operations, such as 1x1 convolutions, emerging convolutions, or periodic convolutions allow at most two of these three advantages. In our experiments on multiple image datasets, we find that Woodbury transformations allow learning of higher-likelihood models than other flow architectures while still enjoying their efficiency advantages.