# Exploring the Links between the Fundamental Lemma and Kernel Regression

arXiv: https://arxiv.org/abs/2403.05368 · HTML: https://mlsanity.com/paper/2403.05368/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 26 references that are in the map.
- First posted on arXiv 8 Mar 2024.

## Builds on (most cited first, 11 of 11)

- [Feedback Linearization based on Gaussian Processes with event-triggered Online Learning](https://mlsanity.com/paper/1911.06565/) (2019)
- [A trajectory-based framework for data-driven system analysis and control](https://mlsanity.com/paper/1903.10723/) (2019)
- [Robust and Kernelized Data-Enabled Predictive Control for Nonlinear Systems](https://mlsanity.com/paper/2206.01866/) (2022)
- [An analytic comparison of regularization methods for Gaussian Processes](https://mlsanity.com/paper/1602.00853/) (2016)
- [Nonlinear Data-Enabled Prediction and Control](https://mlsanity.com/paper/2101.03187/) (2021)
- [Basis functions nonlinear data-enabled predictive control: Consistent and computationally efficient formulations](https://mlsanity.com/paper/2311.05360/) (2023)
- [Data-Driven Internal Model Control of Second-Order Discrete Volterra Systems](https://mlsanity.com/paper/2003.14158/) (2020)
- [Data-based Control of Feedback Linearizable Systems](https://mlsanity.com/paper/2204.01148/) (2022)
- [Data-Based System Analysis and Control of Flat Nonlinear Systems](https://mlsanity.com/paper/2103.02892/) (2021)
- [Data-Driven Optimal Control of Bilinear Systems](https://mlsanity.com/paper/2112.15510/) (2021)
- [Learning controllers from data via kernel-based interpolation](https://mlsanity.com/paper/2304.09577/) (2023)

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

- [From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma](https://mlsanity.com/paper/2508.07494/) (2025)
- [Data-Driven Prediction and Control of Hammerstein-Wiener Systems with Implicit Gaussian Processes](https://mlsanity.com/paper/2501.15849/) (2025)
- [A Behavioral Framework for Data-Driven Modeling of Nonlinear Systems in Vector-Valued Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2605.07052/) (2026)

## Abstract (arXiv)

Generalizations and variations of the fundamental lemma by Willems et al. are an active topic of recent research. In this note, we explore and formalize the links between kernel regression and some known nonlinear extensions of the fundamental lemma. Applying a transformation to the usual linear equation in Hankel matrices, we arrive at an alternative implicit kernel representation of the system trajectories while keeping the requirements on persistency of excitation. We show that this representation is equivalent to the solution of a specific kernel regression problem. We explore the possible structures of the underlying kernel as well as the system classes to which they correspond.