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

TL;DR

This paper compares various regularization techniques for linear system identification, finding that stable spline kernels outperform atomic and nuclear norm-based methods due to their embedding of stability and smoothness information.

Contribution

The paper provides a comprehensive comparison of regularizers, introduces a new class of integral stable spline kernels, and demonstrates their superior performance in practical scenarios.

Findings

Stable spline kernels outperform atomic and nuclear norm methods.

New integral stable spline kernels improve estimation accuracy.

Proposed estimators outperform classical prediction error methods.

Abstract

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.