On the Approximation of Toeplitz Operators for Nonparametric $\mathcal{H}_\infty$-norm Estimation
Provides theoretical guarantees for a nonparametric H∞-norm estimation algorithm, but the result is incremental as it refines existing analysis and highlights limitations compared to parametric methods.
The paper proves sharp non-asymptotic bounds on the length of Toeplitz truncation needed for ε-additive approximation of the H∞-norm of a stable SISO LTI system from noisy data, and shows a family of FIR filters where the power method requires arbitrarily more timesteps than parametric least-squares identification.
Given a stable SISO LTI system $G$, we investigate the problem of estimating the $\mathcal{H}_\infty$-norm of $G$, denoted $||G||_\infty$, when $G$ is only accessible via noisy observations. Wahlberg et al. recently proposed a nonparametric algorithm based on the power method for estimating the top eigenvalue of a matrix. In particular, by applying a clever time-reversal trick, Wahlberg et al. implement the power method on the top left $n \times n$ corner $T_n$ of the Toeplitz (convolution) operator associated with $G$. In this paper, we prove sharp non-asymptotic bounds on the necessary length $n$ needed so that $||T_n||$ is an $\varepsilon$-additive approximation of $||G||_\infty$. Furthermore, in the process of demonstrating the sharpness of our bounds, we construct a simple family of finite impulse response (FIR) filters where the number of timesteps needed for the power method is arbitrarily worse than the number of timesteps needed for parametric FIR identification via least-squares to achieve the same $\varepsilon$-additive approximation.