[Submitted on 16 Aug 2025 (v1), last revised 19 Oct 2026 (this version, v2)]
Abstract:We study the fundamental problem of one-step prediction of a marginally stable unknown nonlinear dynamical system. We describe an algorithm for this problem, based on the technique of spectral filtering, which learns a mapping from past observations to the next based on a spectral representation of the system. Using techniques from online convex optimization, we prove vanishing prediction error for any nonexpansive nonlinear dynamical system with finitely many marginally stable modes, with rates governed by a novel quantitative control-theoretic notion of learnability. The main technical component of our method is a new spectral filtering algorithm for linear dynamical systems, which incorporates past observations and applies to general noisy and marginally stable systems. This generalizes the original spectral filtering algorithm to both asymmetric dynamics as well as incorporating noise correction, and is of independent interest.
Submission history
From: Evan Dogariu [view email]
[v1] Sat, 16 Aug 2025 09:14:47 UTC (4,365 KB)
[v2] Mon, 19 Oct 2026 07:15:09 UTC (3,829 KB)
