# How Implicit Regularization of ReLU Neural Networks Characterizes the Learned Function -- Part I: the 1-D Case of Two Layers with Random First Layer

arXiv: https://arxiv.org/abs/1911.02903 · HTML: https://mlsanity.com/paper/1911.02903/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Nov 2019.

## Built on by (most cited first, 5 of 5)

- [Universal approximation property of Banach space-valued random feature models including random neural networks](https://mlsanity.com/paper/2312.08410/) (2023)
- [RHINO: Regularizing the Hash-based Implicit Neural Representation](https://mlsanity.com/paper/2309.12642/) (2023)
- [Extending Path-Dependent NJ-ODEs to Noisy Observations and a Dependent Observation Framework](https://mlsanity.com/paper/2307.13147/) (2023)
- [How Inverse Conditional Flows Can Serve as a Substitute for Distributional Regression](https://mlsanity.com/paper/2405.05429/) (2024)
- [Gradient Descent as a Shrinkage Operator for Spectral Bias](https://mlsanity.com/paper/2504.18207/) (2025)

## Abstract (arXiv)

In this paper, we consider one dimensional (shallow) ReLU neural networks in which weights are chosen randomly and only the terminal layer is trained. First, we mathematically show that for such networks L2-regularized regression corresponds in function space to regularizing the estimate's second derivative for fairly general loss functionals. For least squares regression, we show that the trained network converges to the smooth spline interpolation of the training data as the number of hidden nodes tends to infinity. Moreover, we derive a novel correspondence between the early stopped gradient descent (without any explicit regularization of the weights) and the smoothing spline regression.