# Sufficient Conditions for Persistency of Excitation with Step and ReLU Activation Functions

arXiv: https://arxiv.org/abs/2209.06286 · HTML: https://mlsanity.com/paper/2209.06286/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Sep 2022.

## Built on by (most cited first, 3 of 3)

- [Approximation with Random Shallow ReLU Networks with Applications to Model Reference Adaptive Control](https://mlsanity.com/paper/2403.17142/) (2024)
- [Evaluating Time-Series Training Dataset through Lens of Spectrum in Deep State Space Models](https://mlsanity.com/paper/2408.16261/) (2024)
- [Function Approximation with Randomly Initialized Neural Networks for Approximate Model Reference Adaptive Control](https://mlsanity.com/paper/2303.16251/) (2023)

## Abstract (arXiv)

This paper defines geometric criteria which are then used to establish sufficient conditions for persistency of excitation with vector functions constructed from single hidden-layer neural networks with step or ReLU activation functions. We show that these conditions hold when employing reference system tracking, as is commonly done in adaptive control. We demonstrate the results numerically on a system with linearly parameterized activations of this type and show that the parameter estimates converge to the true values with the sufficient conditions met.