From STLS to Projection-based Dictionary Selection in Sparse Regression for System Identification

TL;DR

This paper introduces a score-guided library selection method for sparse regression in system identification, improving accuracy and interpretability of SINDy algorithms through theoretical analysis and numerical validation.

Contribution

It proposes a novel score-guided dictionary selection strategy for STLS, with theoretical insights and demonstrated benefits in dynamical system modeling.

Findings

Score-based screening improves model accuracy.

Enhanced interpretability of identified systems.

Method effective for ordinary and partial differential equations.

Abstract

In this work, we revisit dictionary-based sparse regression, in particular, Sequential Threshold Least Squares (STLS), and propose a score-guided library selection to provide practical guidance for data-driven modeling, with emphasis on SINDy-type algorithms. STLS is an algorithm to solve the $\ell_0$ sparse least-squares problem, which relies on splitting to efficiently solve the least-squares portion while handling the sparse term via proximal methods. It produces coefficient vectors whose components depend on both the projected reconstruction errors, here referred to as the scores, and the mutual coherence of dictionary terms. The first contribution of this work is a theoretical analysis of the score and dictionary-selection strategy. This could be understood in both the original and weak SINDy regime. Second, numerical experiments on ordinary and partial differential equations highlight the effectiveness of score-based screening, improving both accuracy and interpretability in dynamical system identification. These results suggest that integrating score-guided methods to refine the dictionary more accurately may help SINDy users in some cases to enhance their robustness for data-driven discovery of governing equations.