# An exact mapping between the Variational Renormalization Group and Deep Learning

arXiv: https://arxiv.org/abs/1410.3831 · HTML: https://mlsanity.com/paper/1410.3831/

## Key facts

- Cited by 56 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Oct 2014.

## Built on by (most cited first, 30 of 56)

- [How Compositional Generalization and Creativity Improve as Diffusion Models are Trained](https://mlsanity.com/paper/2502.12089/) (2025)
- [Nonperturbative renormalization for the neural network-QFT correspondence](https://mlsanity.com/paper/2108.01403/) (2021)
- [Renormalizing Diffusion Models](https://mlsanity.com/paper/2308.12355/) (2023)
- [Towards quantifying information flows: relative entropy in deep neural networks and the renormalization group](https://mlsanity.com/paper/2107.06898/) (2021)
- [Bayesian Renormalization](https://mlsanity.com/paper/2305.10491/) (2023)
- [Tensor networks and efficient descriptions of classical data](https://mlsanity.com/paper/2103.06872/) (2021)
- [Wavelet Conditional Renormalization Group](https://mlsanity.com/paper/2207.04941/) (2022)
- [Dense Hebbian neural networks: a replica symmetric picture of supervised learning](https://mlsanity.com/paper/2212.00606/) (2022)
- [Generalization Dynamics of Linear Diffusion Models](https://mlsanity.com/paper/2505.24769/) (2025)
- [Universality of Winning Tickets: A Renormalization Group Perspective](https://mlsanity.com/paper/2110.03210/) (2021)
- [Wilsonian Renormalization of Neural Network Gaussian Processes](https://mlsanity.com/paper/2405.06008/) (2024)
- [The Autodidactic Universe](https://mlsanity.com/paper/2104.03902/) (2021)
- [Renormalization in the neural network-quantum field theory correspondence](https://mlsanity.com/paper/2212.11811/) (2022)
- [RGMem: Renormalization Group-inspired Memory Evolution for Language Agents](https://mlsanity.com/paper/2510.16392/) (2025)
- [Entropy Regularized Reinforcement Learning Using Large Deviation Theory](https://mlsanity.com/paper/2106.03931/) (2021)
- [Multi-Scale Simulation of Complex Systems: A Perspective of Integrating Knowledge and Data](https://mlsanity.com/paper/2306.10275/) (2023)
- [Deep Learning of Compositional Targets with Hierarchical Spectral Methods](https://mlsanity.com/paper/2602.10867/) (2026)
- [Engineering flexible machine learning systems by traversing functionally-invariant paths](https://mlsanity.com/paper/2205.00334/) (2022)
- [Towards Worst-Case Guarantees with Scale-Aware Interpretability](https://mlsanity.com/paper/2602.05184/) (2026)
- [Learning ECG Signal Features Without Backpropagation Using Linear Laws](https://mlsanity.com/paper/2307.01930/) (2023)
- [Latent Object Permanence: Topological Phase Transitions, Free-Energy Principles, and Renormalization Group Flows in Deep Transformer Manifolds](https://mlsanity.com/paper/2601.19942/) (2026)
- [Feature extraction of machine learning and phase transition point of Ising model](https://mlsanity.com/paper/2111.11166/) (2021)
- [Iterative Magnitude Pruning as a Renormalisation Group: A Study in The Context of The Lottery Ticket Hypothesis](https://mlsanity.com/paper/2308.03128/) (2023)
- [MS3D: A RG Flow-Based Regularization for GAN Training with Limited Data](https://mlsanity.com/paper/2408.11135/) (2024)
- [Dynamic neuron approach to deep neural networks: Decoupling neurons for renormalization group analysis](https://mlsanity.com/paper/2410.00396/) (2024)
- [A Two-Phase Perspective on Deep Learning Dynamics](https://mlsanity.com/paper/2504.12700/) (2025)
- [Borrowed Geometry: Cross-Distribution Head-Importance Fingerprints of Frozen Pretrained Gemma 4 31B](https://mlsanity.com/paper/2605.00333/) (2026)
- [Rank Collapse, Fixed Points, and the Renormalization Group Structure of MLP Residual Networks](https://mlsanity.com/paper/2606.10324/) (2026)
- [Broken Ergodicity and the Violation of the Fluctuation-Dissipation Theorem Lead to Generalization Beyond Overfitting in Machine Learning](https://mlsanity.com/paper/2607.04135/) (2026)
- [The Physics of Machine Learning: An Intuitive Introduction for the Physical Scientist](https://mlsanity.com/paper/2112.00851/) (2021)

## Abstract (arXiv)

Deep learning is a broad set of techniques that uses multiple layers of representation to automatically learn relevant features directly from structured data. Recently, such techniques have yielded record-breaking results on a diverse set of difficult machine learning tasks in computer vision, speech recognition, and natural language processing. Despite the enormous success of deep learning, relatively little is understood theoretically about why these techniques are so successful at feature learning and compression. Here, we show that deep learning is intimately related to one of the most important and successful techniques in theoretical physics, the renormalization group (RG). RG is an iterative coarse-graining scheme that allows for the extraction of relevant features (i.e. operators) as a physical system is examined at different length scales. We construct an exact mapping from the variational renormalization group, first introduced by Kadanoff, and deep learning architectures based on Restricted Boltzmann Machines (RBMs). We illustrate these ideas using the nearest-neighbor Ising Model in one and two-dimensions. Our results suggests that deep learning algorithms may be employing a generalized RG-like scheme to learn relevant features from data.