Spectral Filtering for Complex Linear Dynamical Systems

TL;DR

This paper introduces a spectral filtering approach using the Slepian basis to learn complex-valued linear dynamical systems with sector-bounded spectra, enabling dimension-free prediction bounds.

Contribution

It presents a novel spectral filtering method for CLDS with spectrum in a sector, achieving learnability results independent of ambient dimension.

Findings

Learnability governed by an effective dimension independent of state dimension

Dimension-free regret bounds for sequence prediction in CLDS

Spectral filtering based on the Slepian basis effectively captures oscillatory dynamics

Abstract

We study the problem of learning complex-valued linear dynamical systems (CLDS) with sector-bounded spectrum. This class captures oscillatory and long-memory dynamics arising in signal processing, structured state space models, and quantum systems. We introduce a spectral filtering method based on the Slepian basis and show that learnability is governed by an effective dimension independent of the ambient state dimension. As a consequence, we obtain dimension-free regret bounds for sequence prediction in CLDS with spectrum contained in a sector of the unit disk.