# Learning Fractals by Gradient Descent

arXiv: https://arxiv.org/abs/2303.12722 · HTML: https://mlsanity.com/paper/2303.12722/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 59 references that are in the map.
- First posted on arXiv 14 Mar 2023.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [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)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [GAN Inversion: A Survey](https://mlsanity.com/paper/2101.05278/) (2021)
- [Generative Visual Manipulation on the Natural Image Manifold](https://mlsanity.com/paper/1609.03552/) (2016)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [FractalNet: Ultra-Deep Neural Networks without Residuals](https://mlsanity.com/paper/1605.07648/) (2016)
- [PixMix: Dreamlike Pictures Comprehensively Improve Safety Measures](https://mlsanity.com/paper/2112.05135/) (2021)
- [Pre-training without Natural Images](https://mlsanity.com/paper/2101.08515/) (2021)
- [Learning to See by Looking at Noise](https://mlsanity.com/paper/2106.05963/) (2021)
- [End-to-End Pseudo-LiDAR for Image-Based 3D Object Detection](https://mlsanity.com/paper/2004.03080/) (2020)
- [Can Vision Transformers Learn without Natural Images?](https://mlsanity.com/paper/2103.13023/) (2021)
- [Improving Fractal Pre-training](https://mlsanity.com/paper/2110.03091/) (2021)
- [Fractal Structure and Generalization Properties of Stochastic Optimization Algorithms](https://mlsanity.com/paper/2106.04881/) (2021)
- [Expression of Fractals Through Neural Network Functions](https://mlsanity.com/paper/1905.11345/) (2019)
- [Self-Similarity Priors: Neural Collages as Differentiable Fractal Representations](https://mlsanity.com/paper/2204.07673/) (2022)
- [Fractal Pyramid Networks](https://mlsanity.com/paper/2106.14694/) (2021)
- [Beyond Flatland: Pre-training with a Strong 3D Inductive Bias](https://mlsanity.com/paper/2112.00113/) (2021)

## Built on by (most cited first, 2 of 2)

- [Learning with Mandelbrot and Julia](https://mlsanity.com/paper/2509.00903/) (2025)
- [Amortized Set Prediction for Inverse IFS Reconstruction from Density Maps](https://mlsanity.com/paper/2608.24175/) (2026)

## Abstract (arXiv)

Fractals are geometric shapes that can display complex and self-similar patterns found in nature (e.g., clouds and plants). Recent works in visual recognition have leveraged this property to create random fractal images for model pre-training. In this paper, we study the inverse problem -- given a target image (not necessarily a fractal), we aim to generate a fractal image that looks like it. We propose a novel approach that learns the parameters underlying a fractal image via gradient descent. We show that our approach can find fractal parameters of high visual quality and be compatible with different loss functions, opening up several potentials, e.g., learning fractals for downstream tasks, scientific understanding, etc.