# Emerging Convolutions for Generative Normalizing Flows

arXiv: https://arxiv.org/abs/1901.11137 · HTML: https://mlsanity.com/paper/1901.11137/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jan 2019.

## Built on by (most cited first, 30 of 31)

- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [A Comprehensive Survey of AI-Generated Content (AIGC): A History of Generative AI from GAN to ChatGPT](https://mlsanity.com/paper/2303.04226/) (2023)
- [Graph-Augmented Normalizing Flows for Anomaly Detection of Multiple Time Series](https://mlsanity.com/paper/2202.07857/) (2022)
- [Implicit Normalizing Flows](https://mlsanity.com/paper/2103.09527/) (2021)
- [Maximum Entropy Reinforcement Learning via Energy-Based Normalizing Flow](https://mlsanity.com/paper/2405.13629/) (2024)
- [MolGrow: A Graph Normalizing Flow for Hierarchical Molecular Generation](https://mlsanity.com/paper/2106.05856/) (2021)
- [Generative Flows with Invertible Attentions](https://mlsanity.com/paper/2106.03959/) (2021)
- [ButterflyFlow: Building Invertible Layers with Butterfly Matrices](https://mlsanity.com/paper/2209.13774/) (2022)
- [Jet: A Modern Transformer-Based Normalizing Flow](https://mlsanity.com/paper/2412.15129/) (2024)
- [Training Energy-Based Normalizing Flow with Score-Matching Objectives](https://mlsanity.com/paper/2305.15267/) (2023)
- [Invertible Attention](https://mlsanity.com/paper/2106.09003/) (2021)
- [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)
- [CInC Flow: Characterizable Invertible 3x3 Convolution](https://mlsanity.com/paper/2107.01358/) (2021)
- [Funnels: Exact maximum likelihood with dimensionality reduction](https://mlsanity.com/paper/2112.08069/) (2021)
- [Transformer-Based Speech Synthesizer Attribution in an Open Set Scenario](https://mlsanity.com/paper/2210.07546/) (2022)
- [Flow-based Spatio-Temporal Structured Prediction of Motion Dynamics](https://mlsanity.com/paper/2104.04391/) (2021)
- [Segment-Level Road Obstacle Detection Using Visual Foundation Model Priors and Likelihood Ratios](https://mlsanity.com/paper/2412.05707/) (2024)
- [Orthonormal Convolutions for the Rotation Based Iterative Gaussianization](https://mlsanity.com/paper/2206.03860/) (2022)
- [FInC Flow: Fast and Invertible k × k Convolutions for Normalizing Flows](https://mlsanity.com/paper/2301.09266/) (2023)
- [PixelPyramids: Exact Inference Models from Lossless Image Pyramids](https://mlsanity.com/paper/2110.08787/) (2021)
- [Characteristic Neural Ordinary Differential Equations](https://mlsanity.com/paper/2111.13207/) (2021)
- [VeriFlow: Modeling Distributions for Neural Network Verification](https://mlsanity.com/paper/2406.14265/) (2024)
- [CDFlow: Building Invertible Layers with Circulant and Diagonal Matrices](https://mlsanity.com/paper/2510.25323/) (2025)
- [Delta-GAN-Encoder: Encoding Semantic Changes for Explicit Image Editing, using Few Synthetic Samples](https://mlsanity.com/paper/2111.08419/) (2021)
- [Invertible Convolution with Symmetric Paddings](https://mlsanity.com/paper/2303.17361/) (2023)
- [MixerFlow: MLP-Mixer meets Normalising Flows](https://mlsanity.com/paper/2310.16777/) (2023)
- [Parallel Backpropagation for Inverse of a Convolution with Application to Normalizing Flows](https://mlsanity.com/paper/2410.14634/) (2024)
- [On Uniformly Scaling Flows: A Density-Aligned Approach to Deep One-Class Classification](https://mlsanity.com/paper/2510.09452/) (2025)
- [MAGIC-Flow: Multiscale Adaptive Conditional Flows for Generation and Interpretable Classification](https://mlsanity.com/paper/2510.22070/) (2025)

## Abstract (arXiv)

Generative flows are attractive because they admit exact likelihood optimization and efficient image synthesis. Recently, Kingma & Dhariwal (2018) demonstrated with Glow that generative flows are capable of generating high quality images. We generalize the 1 x 1 convolutions proposed in Glow to invertible d x d convolutions, which are more flexible since they operate on both channel and spatial axes. We propose two methods to produce invertible convolutions that have receptive fields identical to standard convolutions: Emerging convolutions are obtained by chaining specific autoregressive convolutions, and periodic convolutions are decoupled in the frequency domain. Our experiments show that the flexibility of d x d convolutions significantly improves the performance of generative flow models on galaxy images, CIFAR10 and ImageNet.