# How do Minimum-Norm Shallow Denoisers Look in Function Space?

arXiv: https://arxiv.org/abs/2311.06748 · HTML: https://mlsanity.com/paper/2311.06748/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 31 references that are in the map.
- First posted on arXiv 12 Nov 2023.

## Builds on (most cited first, 18 of 18)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [How do infinite width bounded norm networks look in function space?](https://mlsanity.com/paper/1902.05040/) (2019)
- [Spontaneous Symmetry Breaking in Generative Diffusion Models](https://mlsanity.com/paper/2305.19693/) (2023)
- [Image Denoising: The Deep Learning Revolution and Beyond -- A Survey Paper --](https://mlsanity.com/paper/2301.03362/) (2023)
- [What Kinds of Functions do Deep Neural Networks Learn? Insights from Variational Spline Theory](https://mlsanity.com/paper/2105.03361/) (2021)
- [Convex Geometry and Duality of Over-parameterized Neural Networks](https://mlsanity.com/paper/2002.11219/) (2020)
- [Implicit Bias of Large Depth Networks: a Notion of Rank for Nonlinear Functions](https://mlsanity.com/paper/2209.15055/) (2022)
- [Penalising the biases in norm regularisation enforces sparsity](https://mlsanity.com/paper/2303.01353/) (2023)
- [Feature Learning in L2-regularized DNNs: Attraction/Repulsion and Sparsity](https://mlsanity.com/paper/2205.15809/) (2022)
- [Convex Regularization Behind Neural Reconstruction](https://mlsanity.com/paper/2012.05169/) (2020)
- [The Implicit Bias of Minima Stability in Multivariate Shallow ReLU Networks](https://mlsanity.com/paper/2306.17499/) (2023)
- [Ridgeless Interpolation with Shallow ReLU Networks in 1D is Nearest Neighbor Curvature Extrapolation and Provably Generalizes on Lipschitz Functions](https://mlsanity.com/paper/2109.12960/) (2021)

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

- [Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structure](https://mlsanity.com/paper/2410.24060/) (2024)
- [How Uniform Random Weights Induce Non-uniform Bias: Typical Interpolating Neural Networks Generalize with Narrow Teachers](https://mlsanity.com/paper/2402.06323/) (2024)
- [Taking a Big Step: Large Learning Rates in Denoising Score Matching Prevent Memorization](https://mlsanity.com/paper/2502.03435/) (2025)
- [Generalization of Diffusion Models Arises with a Balanced Representation Space](https://mlsanity.com/paper/2512.20963/) (2025)
- [Analyzing Neural Network-Based Generative Diffusion Models through Convex Optimization](https://mlsanity.com/paper/2402.01965/) (2024)
- [When Diffusion Models Memorize: Inductive Biases in Probability Flow of Minimum-Norm Shallow Neural Nets](https://mlsanity.com/paper/2506.19031/) (2025)
- [ReLUs Are Sufficient for Learning Implicit Neural Representations](https://mlsanity.com/paper/2406.02529/) (2024)
- [Two Calm Ends and the Wild Middle: A Geometric Picture of Memorization in Diffusion Models](https://mlsanity.com/paper/2602.17846/) (2026)
- [The Effects of Multi-Task Learning on ReLU Neural Network Functions](https://mlsanity.com/paper/2410.21696/) (2024)

## Abstract (arXiv)

Neural network (NN) denoisers are an essential building block in many common tasks, ranging from image reconstruction to image generation. However, the success of these models is not well understood from a theoretical perspective. In this paper, we aim to characterize the functions realized by shallow ReLU NN denoisers -- in the common theoretical setting of interpolation (i.e., zero training loss) with a minimal representation cost (i.e., minimal ℓ2 norm weights). First, for univariate data, we derive a closed form for the NN denoiser function, find it is contractive toward the clean data points, and prove it generalizes better than the empirical MMSE estimator at a low noise level. Next, for multivariate data, we find the NN denoiser functions in a closed form under various geometric assumptions on the training data: data contained in a low-dimensional subspace, data contained in a union of one-sided rays, or several types of simplexes. These functions decompose into a sum of simple rank-one piecewise linear interpolations aligned with edges and/or faces connecting training samples. We empirically verify this alignment phenomenon on synthetic data and real images.