Kolmogorov $n$-widths for linear dynamical systems

TL;DR

This paper reveals a direct connection between Kolmogorov n-widths and Hankel singular values in linear time-invariant systems, providing new insights into model reduction techniques and their theoretical underpinnings.

Contribution

It establishes a precise relationship between Kolmogorov n-widths and Hankel singular values for LTI systems, and links active subspaces to Kolmogorov n-widths.

Findings

Kolmogorov n-width equals the (n+1)st Hankel singular value for LTI systems.

A lower bound for the Kolmogorov n-width in parametric LTI systems is derived.

Active subspaces are shown to be dual to the minimizing subspace for Kolmogorov n-width.

Abstract

Kolmogorov $n$-widths and Hankel singular values are two commonly used concepts in model reduction. Here we show that for the special case of linear time-invariant dynamical (LTI) systems, these two concepts are directly connected. More specifically, the greedy search applied to the Hankel operator of an LTI system resembles the minimizing subspace for the Kolmogorov n-width and the Kolmogorov $n$-width of an LTI system equals its $(n+1)st$ Hankel singular value once the subspaces are appropriately defined. We also establish a lower bound for the Kolmorogov $n$-width for parametric LTI systems and illustrate that the method of active subspaces can be viewed as the dual concept to the minimizing subspace for the Kolmogorov $n$-width.