# Regularized linear system identification using atomic, nuclear and kernel-based norms: the role of the stability constraint

arXiv: https://arxiv.org/abs/1507.00564 · HTML: https://mlsanity.com/paper/1507.00564/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jul 2015.

## Built on by (most cited first, 11 of 11)

- [Transformers as Algorithms: Generalization and Stability in In-context Learning](https://mlsanity.com/paper/2301.07067/) (2023)
- [Deep networks for system identification: a Survey](https://mlsanity.com/paper/2301.12832/) (2023)
- [Model order reduction of deep structured state-space models: A system-theoretic approach](https://mlsanity.com/paper/2403.14833/) (2024)
- [Nonconvex Linear System Identification with Minimal State Representation](https://mlsanity.com/paper/2504.18791/) (2025)
- [Hankel Singular Value Regularization for Highly Compressible State Space Models](https://mlsanity.com/paper/2510.22951/) (2025)
- [Likelihood-based generalization of Markov parameter estimation and multiple shooting objectives in system identification](https://mlsanity.com/paper/2212.13902/) (2022)
- [Infinite-Dimensional Sparse Learning in Linear System Identification](https://mlsanity.com/paper/2203.14731/) (2022)
- [Robust Bayesian Subspace Identification for Small Data Sets](https://mlsanity.com/paper/2212.14132/) (2022)
- [Error Bounds for Kernel-Based Linear System Identification with Unknown Hyperparameters](https://mlsanity.com/paper/2303.09842/) (2023)
- [Frequency Response Identification of Low-Order Systems: Finite-Sample Analysis](https://mlsanity.com/paper/2508.17142/) (2025)
- [XMSE-Aware Adaptive Empirical Bayes Estimation](https://mlsanity.com/paper/2606.26975/) (2026)

## Abstract (arXiv)

Inspired by ideas taken from the machine learning literature, new regularization techniques have been recently introduced in linear system identification. In particular, all the adopted estimators solve a regularized least squares problem, differing in the nature of the penalty term assigned to the impulse response. Popular choices include atomic and nuclear norms (applied to Hankel matrices) as well as norms induced by the so called stable spline kernels. In this paper, a comparative study of estimators based on these different types of regularizers is reported. Our findings reveal that stable spline kernels outperform approaches based on atomic and nuclear norms since they suitably embed information on impulse response stability and smoothness. This point is illustrated using the Bayesian interpretation of regularization. We also design a new class of regularizers defined by "integral" versions of stable spline/TC kernels. Under quite realistic experimental conditions, the new estimators outperform classical prediction error methods also when the latter are equipped with an oracle for model order selection.