Kolmogorov $n$-widths for linear dynamical systems
For researchers in model reduction of LTI systems, this work provides theoretical unification of two key concepts, but it is incremental as it builds on known ideas.
The paper establishes a direct connection between Kolmogorov n-widths and Hankel singular values for linear time-invariant (LTI) systems, showing that the greedy search on the Hankel operator yields the minimizing subspace for the Kolmogorov n-width and that the n-width equals the (n+1)st Hankel singular value. It also provides a lower bound for parametric LTI systems and relates active subspaces to the dual concept.
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.