On the Approximation of Toeplitz Operators for Nonparametric $\mathcal{H}_{\infty}$-norm Estimation

Given a stable SISO LTI system <tex>$G$</tex>, we investigate the problem of estimating the <tex>$\mathcal{H}_{\infty}$</tex>-norm of <tex>$G$</tex>, denoted <tex>$\Vert G\Vert_{\infty}$</tex>, when <tex>$G$</tex> is only accessible via noisy observations. Wahlberg et al. [1] 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 <tex>$n\times n$</tex> corner <tex>$T_{n}$</tex> of the Toeplitz (convolution) operator associated to <tex>$G$</tex>. In this paper, we prove sharp non-asymptotic bounds on the necessary length <tex>$n$</tex> needed so that <tex>$\Vert T_{n}\Vert$</tex> is an <tex>$\varepsilon$</tex>-additive approximation of <tex>$\Vert G\Vert_{\infty}$</tex>. 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 <tex>$\varepsilon$</tex>-additive approximation.