# Wavelet Flow: Fast Training of High Resolution Normalizing Flows

arXiv: https://arxiv.org/abs/2010.13821 · HTML: https://mlsanity.com/paper/2010.13821/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Oct 2020.

## Built on by (most cited first, 30 of 33)

- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Wavelet Score-Based Generative Modeling](https://mlsanity.com/paper/2208.05003/) (2022)
- [Stochastic Image-to-Video Synthesis using cINNs](https://mlsanity.com/paper/2105.04551/) (2021)
- [Densely connected normalizing flows](https://mlsanity.com/paper/2106.04627/) (2021)
- [Fractal Generative Models](https://mlsanity.com/paper/2502.17437/) (2025)
- [PatchNR: Learning from Very Few Images by Patch Normalizing Flow Regularization](https://mlsanity.com/paper/2205.12021/) (2022)
- [Learning multi-scale local conditional probability models of images](https://mlsanity.com/paper/2303.02984/) (2023)
- [Lipschitz-Guided Design of Interpolation Schedules in Generative Models](https://mlsanity.com/paper/2509.01629/) (2025)
- [Conditional Injective Flows for Bayesian Imaging](https://mlsanity.com/paper/2204.07664/) (2022)
- [A Review of Change of Variable Formulas for Generative Modeling](https://mlsanity.com/paper/2308.02652/) (2023)
- [Multiscale Flow for Robust and Optimal Cosmological Analysis](https://mlsanity.com/paper/2306.04689/) (2023)
- [TriNeRFLet: A Wavelet Based Triplane NeRF Representation](https://mlsanity.com/paper/2401.06191/) (2024)
- [Learning Many-to-Many Mapping for Unpaired Real-World Image Super-resolution and Downscaling](https://mlsanity.com/paper/2310.04964/) (2023)
- [Learning a Deep Color Difference Metric for Photographic Images](https://mlsanity.com/paper/2303.14964/) (2023)
- [MolHF: A Hierarchical Normalizing Flow for Molecular Graph Generation](https://mlsanity.com/paper/2305.08457/) (2023)
- [Ship in Sight: Diffusion Models for Ship-Image Super Resolution](https://mlsanity.com/paper/2403.18370/) (2024)
- [PAGER: Progressive Attribute-Guided Extendable Robust Image Generation](https://mlsanity.com/paper/2206.00162/) (2022)
- [Entropy-Informed Weighting Channel Normalizing Flow for Deep Generative Models](https://mlsanity.com/paper/2407.04958/) (2024)
- [Scale-Adaptive Generative Flows for Multiscale Scientific Data](https://mlsanity.com/paper/2509.02971/) (2025)
- [FlowLet: Conditional 3D Brain MRI Synthesis using Wavelet Flow Matching](https://mlsanity.com/paper/2601.05212/) (2026)
- [CAFLOW: Conditional Autoregressive Flows](https://mlsanity.com/paper/2106.02531/) (2021)
- [Multi-Resolution Continuous Normalizing Flows](https://mlsanity.com/paper/2106.08462/) (2021)
- [Learning Non-linear Wavelet Transformation via Normalizing Flow](https://mlsanity.com/paper/2101.11306/) (2021)
- [Efficient Out-of-Distribution Detection of Melanoma with Wavelet-based Normalizing Flows](https://mlsanity.com/paper/2208.04639/) (2022)
- [Conditionally Strongly Log-Concave Generative Models](https://mlsanity.com/paper/2306.00181/) (2023)
- [Fast light-field 3D microscopy with out-of-distribution detection and adaptation through Conditional Normalizing Flows](https://mlsanity.com/paper/2306.06408/) (2023)
- [Analyzing Generative Models by Manifold Entropic Metrics](https://mlsanity.com/paper/2410.19426/) (2024)
- [Likelihood Training of Cascaded Diffusion Models via Hierarchical Volume-preserving Maps](https://mlsanity.com/paper/2501.06999/) (2025)
- [Spectral Dictionary Learning for Generative Image Modeling](https://mlsanity.com/paper/2504.17804/) (2025)
- [Wavelet Flow For Extragalactic Foreground Simulations](https://mlsanity.com/paper/2505.21220/) (2025)

## Abstract (arXiv)

Normalizing flows are a class of probabilistic generative models which allow for both fast density computation and efficient sampling and are effective at modelling complex distributions like images. A drawback among current methods is their significant training cost, sometimes requiring months of GPU training time to achieve state-of-the-art results. This paper introduces Wavelet Flow, a multi-scale, normalizing flow architecture based on wavelets. A Wavelet Flow has an explicit representation of signal scale that inherently includes models of lower resolution signals and conditional generation of higher resolution signals, i.e., super resolution. A major advantage of Wavelet Flow is the ability to construct generative models for high resolution data (e.g., 1024 x 1024 images) that are impractical with previous models. Furthermore, Wavelet Flow is competitive with previous normalizing flows in terms of bits per dimension on standard (low resolution) benchmarks while being up to 15x faster to train.