# A Note on the Regularity of Images Generated by Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2204.10588 · HTML: https://mlsanity.com/paper/2204.10588/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 50 references that are in the map.
- First posted on arXiv 22 Apr 2022.

## Builds on (most cited first, 20 of 20)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network](https://mlsanity.com/paper/1609.04802/) (2016)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [FFDNet: Toward a Fast and Flexible Solution for CNN based Image Denoising](https://mlsanity.com/paper/1710.04026/) (2017)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Deep Decoder: Concise Image Representations from Untrained Non-convolutional Networks](https://mlsanity.com/paper/1810.03982/) (2018)
- [Invertible generative models for inverse problems: mitigating representation error and dataset bias](https://mlsanity.com/paper/1905.11672/) (2019)
- [Robust and interpretable blind image denoising via bias-free convolutional neural networks](https://mlsanity.com/paper/1906.05478/) (2019)
- [Understanding Deep Convolutional Networks](https://mlsanity.com/paper/1601.04920/) (2016)
- [Phase Retrieval Under a Generative Prior](https://mlsanity.com/paper/1807.04261/) (2018)
- [GANs for Medical Image Synthesis: An Empirical Study](https://mlsanity.com/paper/2105.05318/) (2021)
- [Automatic Brain Tumor Detection and Segmentation Using U-Net Based Fully Convolutional Networks](https://mlsanity.com/paper/1705.03820/) (2017)
- [Denoising and Regularization via Exploiting the Structural Bias of Convolutional Generators](https://mlsanity.com/paper/1910.14634/) (2019)
- [A Generative Variational Model for Inverse Problems in Imaging](https://mlsanity.com/paper/2104.12630/) (2021)
- [Deep synthesis regularization of inverse problems](https://mlsanity.com/paper/2002.00155/) (2020)
- [A convex variational model for learning convolutional image atoms from incomplete data](https://mlsanity.com/paper/1812.03077/) (2018)

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

- [Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generation](https://mlsanity.com/paper/2303.04772/) (2023)
- [Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces](https://mlsanity.com/paper/2205.14627/) (2022)

## Abstract (arXiv)

The regularity of images generated by convolutional neural networks, such as the U-net, generative networks, or the deep image prior, is analyzed. In a resolution-independent, infinite dimensional setting, it is shown that such images, represented as functions, are always continuous and, in some circumstances, even continuously differentiable, contradicting the widely accepted modeling of sharp edges in images via jump discontinuities. While such statements require an infinite dimensional setting, the connection to (discretized) neural networks used in practice is made by considering the limit as the resolution approaches infinity. As practical consequence, the results of this paper in particular provide analytical evidence that basic L2 regularization of network weights might lead to over-smoothed outputs.