# RG-Flow: A hierarchical and explainable flow model based on renormalization group and sparse prior

arXiv: https://arxiv.org/abs/2010.00029 · HTML: https://mlsanity.com/paper/2010.00029/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Sep 2020.

## Built on by (most cited first, 11 of 11)

- [Multilevel Generative Samplers for Investigating Critical Phenomena](https://mlsanity.com/paper/2503.08918/) (2025)
- [Learning Non-linear Wavelet Transformation via Normalizing Flow](https://mlsanity.com/paper/2101.11306/) (2021)
- [Neural Information Squeezer for Causal Emergence](https://mlsanity.com/paper/2201.10154/) (2022)
- [Holographic-(V)AE: an end-to-end SO(3)-Equivariant (Variational) Autoencoder in Fourier Space](https://mlsanity.com/paper/2209.15567/) (2022)
- [Probabilistic and Semantic Descriptions of Image Manifolds and Their Applications](https://mlsanity.com/paper/2307.02881/) (2023)
- [Differentiable Programming of Isometric Tensor Networks](https://mlsanity.com/paper/2110.03898/) (2021)
- [Learning ECG Signal Features Without Backpropagation Using Linear Laws](https://mlsanity.com/paper/2307.01930/) (2023)
- [Pre-trained Multiple Latent Variable Generative Models are good defenders against Adversarial Attacks](https://mlsanity.com/paper/2412.03453/) (2024)
- [Categorical Representation Learning and RG flow operators for algorithmic classifiers](https://mlsanity.com/paper/2203.07975/) (2022)
- [A Mutual Information Perspective on Multiple Latent Variable Generative Models for Positive View Generation](https://mlsanity.com/paper/2501.13718/) (2025)
- [MotionPyramid: Hierarchical Motion Representation and Residual Interfaces](https://mlsanity.com/paper/2606.20705/) (2026)

## Abstract (arXiv)

Flow-based generative models have become an important class of unsupervised learning approaches. In this work, we incorporate the key ideas of renormalization group (RG) and sparse prior distribution to design a hierarchical flow-based generative model, RG-Flow, which can separate information at different scales of images and extract disentangled representations at each scale. We demonstrate our method on synthetic multi-scale image datasets and the CelebA dataset, showing that the disentangled representations enable semantic manipulation and style mixing of the images at different scales. To visualize the latent representations, we introduce receptive fields for flow-based models and show that the receptive fields of RG-Flow are similar to those of convolutional neural networks. In addition, we replace the widely adopted isotropic Gaussian prior distribution by the sparse Laplacian distribution to further enhance the disentanglement of representations. From a theoretical perspective, our proposed method has O(log L) complexity for inpainting of an image with edge length L, compared to previous generative models with O(L2) complexity.