# Learning Non-linear Wavelet Transformation via Normalizing Flow

arXiv: https://arxiv.org/abs/2101.11306 · HTML: https://mlsanity.com/paper/2101.11306/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 64 references that are in the map.
- First posted on arXiv 27 Jan 2021.

## Builds on (most cited first, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modifications](https://mlsanity.com/paper/1701.05517/) (2017)
- [A Downsampled Variant of ImageNet as an Alternative to the CIFAR datasets](https://mlsanity.com/paper/1707.08819/) (2017)
- [Guided Image Generation with Conditional Invertible Neural Networks](https://mlsanity.com/paper/1907.02392/) (2019)
- [Disentanglement by Nonlinear ICA with General Incompressible-flow Networks (GIN)](https://mlsanity.com/paper/2001.04872/) (2020)
- [Integer Discrete Flows and Lossless Compression](https://mlsanity.com/paper/1905.07376/) (2019)
- [Neural Network Renormalization Group](https://mlsanity.com/paper/1802.02840/) (2018)
- [A-NICE-MC: Adversarial Training for MCMC](https://mlsanity.com/paper/1706.07561/) (2017)
- [IDF++: Analyzing and Improving Integer Discrete Flows for Lossless Compression](https://mlsanity.com/paper/2006.12459/) (2020)
- [Compression with Flows via Local Bits-Back Coding](https://mlsanity.com/paper/1905.08500/) (2019)
- [Wavelet Flow: Fast Training of High Resolution Normalizing Flows](https://mlsanity.com/paper/2010.13821/) (2020)
- [Joint Sub-bands Learning with Clique Structures for Wavelet Domain Super-Resolution](https://mlsanity.com/paper/1809.04508/) (2018)
- [Interpretable Convolutional Filters with SincNet](https://mlsanity.com/paper/1811.09725/) (2018)
- [Neural Canonical Transformation with Symplectic Flows](https://mlsanity.com/paper/1910.00024/) (2019)
- [RG-Flow: A hierarchical and explainable flow model based on renormalization group and sparse prior](https://mlsanity.com/paper/2010.00029/) (2020)
- [Using Wavelets and Spectral Methods to Study Patterns in Image-Classification Datasets](https://mlsanity.com/paper/2006.09879/) (2020)

## Built on by (most cited first, 3 of 3)

- [Wavelet Score-Based Generative Modeling](https://mlsanity.com/paper/2208.05003/) (2022)
- [Learning multi-scale local conditional probability models of images](https://mlsanity.com/paper/2303.02984/) (2023)
- [Differentiable Programming of Isometric Tensor Networks](https://mlsanity.com/paper/2110.03898/) (2021)

## Abstract (arXiv)

Wavelet transformation stands as a cornerstone in modern data analysis and signal processing. Its mathematical essence is an invertible transformation that discerns slow patterns from fast ones in the frequency domain. Such an invertible transformation can be learned by a designed normalizing flow model. With a generalized lifting scheme as coupling layers, a factor-out layer resembling the downsampling, and parameter sharing at different levels of the model, one can train the normalizing flow to filter high-frequency elements at different levels, thus extending traditional linear wavelet transformations to learnable non-linear deep learning models. In this paper, a way of building such flow is proposed, along with a numerical analysis of the learned transformation. Then, we demonstrate the model's ability in image lossless compression, show it can achieve SOTA compression scores while achieving a small model size, substantial generalization ability, and the ability to handle high-dimensional data.