# The Convolution Exponential and Generalized Sylvester Flows

arXiv: https://arxiv.org/abs/2006.01910 · HTML: https://mlsanity.com/paper/2006.01910/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jun 2020.

## Built on by (most cited first, 18 of 18)

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- [Equivariant Finite Normalizing Flows](https://mlsanity.com/paper/2110.08649/) (2021)
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- [Invertible Monotone Operators for Normalizing Flows](https://mlsanity.com/paper/2210.08176/) (2022)
- [Flowification: Everything is a Normalizing Flow](https://mlsanity.com/paper/2205.15209/) (2022)
- [Multi-Resolution Continuous Normalizing Flows](https://mlsanity.com/paper/2106.08462/) (2021)
- [StyleFlow: Disentangle Latent Representations via Normalizing Flow for Unsupervised Text Style Transfer](https://mlsanity.com/paper/2212.09670/) (2022)
- [Normalizing Flows for Hierarchical Bayesian Analysis: A Gravitational Wave Population Study](https://mlsanity.com/paper/2211.09008/) (2022)
- [A Novel Application of Conditional Normalizing Flows: Stellar Age Inference with Gyrochronology](https://mlsanity.com/paper/2307.08753/) (2023)
- [Gradients of Functions of Large Matrices](https://mlsanity.com/paper/2405.17277/) (2024)
- [Multiple Invertible and Partial-Equivariant Function for Latent Vector Transformation to Enhance Disentanglement in VAEs](https://mlsanity.com/paper/2502.03740/) (2025)

## Abstract (arXiv)

This paper introduces a new method to build linear flows, by taking the exponential of a linear transformation. This linear transformation does not need to be invertible itself, and the exponential has the following desirable properties: it is guaranteed to be invertible, its inverse is straightforward to compute and the log Jacobian determinant is equal to the trace of the linear transformation. An important insight is that the exponential can be computed implicitly, which allows the use of convolutional layers. Using this insight, we develop new invertible transformations named convolution exponentials and graph convolution exponentials, which retain the equivariance of their underlying transformations. In addition, we generalize Sylvester Flows and propose Convolutional Sylvester Flows which are based on the generalization and the convolution exponential as basis change. Empirically, we show that the convolution exponential outperforms other linear transformations in generative flows on CIFAR10 and the graph convolution exponential improves the performance of graph normalizing flows. In addition, we show that Convolutional Sylvester Flows improve performance over residual flows as a generative flow model measured in log-likelihood.