# Images as Weight Matrices: Sequential Image Generation Through Synaptic Learning Rules

arXiv: https://arxiv.org/abs/2210.06184 · HTML: https://mlsanity.com/paper/2210.06184/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 53 references that are in the map.
- First posted on arXiv 7 Oct 2022.

## Builds on (most cited first, 30 of 32)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Matching Networks for One Shot Learning](https://mlsanity.com/paper/1606.04080/) (2016)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [Training Generative Adversarial Networks with Limited Data](https://mlsanity.com/paper/2006.06676/) (2020)
- [Alias-Free Generative Adversarial Networks](https://mlsanity.com/paper/2106.12423/) (2021)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Linear Transformers Are Secretly Fast Weight Programmers](https://mlsanity.com/paper/2102.11174/) (2021)
- [PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modifications](https://mlsanity.com/paper/1701.05517/) (2017)
- [Differentiable Augmentation for Data-Efficient GAN Training](https://mlsanity.com/paper/2006.10738/) (2020)
- [Random Feature Attention](https://mlsanity.com/paper/2103.02143/) (2021)
- [Geometric GAN](https://mlsanity.com/paper/1705.02894/) (2017)
- [Towards Faster and Stabilized GAN Training for High-fidelity Few-shot Image Synthesis](https://mlsanity.com/paper/2101.04775/) (2021)
- [High Fidelity Speech Synthesis with Adversarial Networks](https://mlsanity.com/paper/1909.11646/) (2019)
- [Going Beyond Linear Transformers with Recurrent Fast Weight Programmers](https://mlsanity.com/paper/2106.06295/) (2021)
- [Learning Associative Inference Using Fast Weight Memory](https://mlsanity.com/paper/2011.07831/) (2020)
- [The Dual Form of Neural Networks Revisited: Connecting Test Time Predictions to Training Patterns via Spotlights of Attention](https://mlsanity.com/paper/2202.05798/) (2022)
- [A Modern Self-Referential Weight Matrix That Learns to Modify Itself](https://mlsanity.com/paper/2202.05780/) (2022)
- [Training language GANs from Scratch](https://mlsanity.com/paper/1905.09922/) (2019)
- [Generative Adversarial Networks are Special Cases of Artificial Curiosity (1990) and also Closely Related to Predictability Minimization (1991)](https://mlsanity.com/paper/1906.04493/) (2019)

## Built on by (most cited first, 9 of 9)

- [Gated Delta Networks: Improving Mamba2 with Delta Rule](https://mlsanity.com/paper/2412.06464/) (2024)
- [Parallelizing Linear Transformers with the Delta Rule over Sequence Length](https://mlsanity.com/paper/2406.06484/) (2024)
- [Learning to (Learn at Test Time): RNNs with Expressive Hidden States](https://mlsanity.com/paper/2407.04620/) (2024)
- [End-to-End Test-Time Training for Long Context](https://mlsanity.com/paper/2512.23675/) (2025)
- [Practical Computational Power of Linear Transformers and Their Recurrent and Self-Referential Extensions](https://mlsanity.com/paper/2310.16076/) (2023)
- [Fast weight programming and linear transformers: from machine learning to neurobiology](https://mlsanity.com/paper/2508.08435/) (2025)
- [Metalearning Continual Learning Algorithms](https://mlsanity.com/paper/2312.00276/) (2023)
- [Learning to Control Rapidly Changing Synaptic Connections: An Alternative Type of Memory in Sequence Processing Artificial Neural Networks](https://mlsanity.com/paper/2211.09440/) (2022)
- [Who invented deep residual learning?](https://mlsanity.com/paper/2509.24732/) (2025)

## Abstract (arXiv)

Work on fast weight programmers has demonstrated the effectiveness of key/value outer product-based learning rules for sequentially generating a weight matrix (WM) of a neural net (NN) by another NN or itself. However, the weight generation steps are typically not visually interpretable by humans, because the contents stored in the WM of an NN are not. Here we apply the same principle to generate natural images. The resulting fast weight painters (FPAs) learn to execute sequences of delta learning rules to sequentially generate images as sums of outer products of self-invented keys and values, one rank at a time, as if each image was a WM of an NN. We train our FPAs in the generative adversarial networks framework, and evaluate on various image datasets. We show how these generic learning rules can generate images with respectable visual quality without any explicit inductive bias for images. While the performance largely lags behind the one of specialised state-of-the-art image generators, our approach allows for visualising how synaptic learning rules iteratively produce complex connection patterns, yielding human-interpretable meaningful images. Finally, we also show that an additional convolutional U-Net (now popular in diffusion models) at the output of an FPA can learn one-step "denoising" of FPA-generated images to enhance their quality. Our code is public.