On Embeddings and Inverse Embeddings of Input Design for Regularized System Identification

TL;DR

This paper explores new theoretical insights into embeddings and inverse embeddings in regularized system identification, introducing frequency domain and graph-based perspectives to deepen understanding beyond traditional real and frequency domain views.

Contribution

It introduces a general frequency domain inverse embedding, relates time and frequency domain embeddings via graph signal processing, and proposes a novel graph-induced embedding approach.

Findings

Established a general frequency domain inverse embedding (FDIE).

Connected TDIE and FDIE through graph signal processing.

Proposed a new graph-induced embedding and inverse.

Abstract

Input design is an important problem for system identification and has been well studied for the classical system identification, i.e., the maximum likelihood/prediction error method. For the emerging regularized system identification, the study on input design has just started, and it is often formulated as a non-convex optimization problem that minimizes a scalar measure of the Bayesian mean squared error matrix subject to certain constraints, and the state-of-art method is the so-called quadratic mapping and inverse embedding (QMIE) method, where a time domain inverse embedding (TDIE) is proposed to find the inverse of the quadratic mapping. In this paper, we report some new results on the embeddings/inverse embeddings of the QMIE method. Firstly, we present a general result on the frequency domain inverse embedding (FDIE) that is to find the inverse of the quadratic mapping described by the discrete-time Fourier transform. Then we show the relation between the TDIE and the FDIE from a graph signal processing perspective. Finally, motivated by this perspective, we further propose a graph induced embedding and its inverse, which include the previously introduced embeddings as special cases. This deepens the understanding of input design from a new viewpoint beyond the real domain and the frequency domain viewpoints.