Spectral Distillation: From Nonlinear Dynamics to Linear State-Space Models

2026-08-05

AI Summary

This work addresses the challenge of learning compact linear state-space representations of nonlinear dynamical systems from observational data, circumventing the need to solve non-convex system identification problems directly. The authors propose a two-stage, provably correct pipeline: first, an implicit spectral predictor is learned via Observation Spectral Filtering (OSF), a convex optimization method; second, this predictor is distilled into an explicit linear dynamical system (LDS) through a spectral-to-LDS distillation procedure. This approach yields the first end-to-end provable linearization of nonlinear systems, with error bounds that depend only on observation complexity rather than latent dimensionality and exhibit exponentially small distillation error. Empirical results demonstrate that the resulting compact LDS predictors match or outperform baseline models trained directly on standard LDS benchmarks and MuJoCo behavioral cloning tasks.

Abstract

Can nonlinear dynamical systems be learned through a compact linear state-space representation, without directly solving a non-convex system-identification problem? We give a provable pipeline for doing so. Starting from observations of an unknown nonlinear dynamical system, we first learn an implicit spectral predictor using Observation Spectral Filtering (OSF), a convex method that competes with the best linear observer for the system. We then apply spectral-to-LDS distillation to convert this predictor into an explicit recurrent linear dynamical system. Our main theorem shows that the average prediction error of the distilled LDS decomposes into an exponentially-small distillation term and the OSF learning term governed by the Luenberger complexity of the best observer. The guarantee is dimension-free: it depends on observer complexity rather than on the latent dimension needed to represent the nonlinear system. To our knowledge, this yields the first end-to-end provable method for extracting a best-in-hindsight LDS representation of nonlinear dynamics through convex learning followed by provable distillation. Experiments on linear LDS benchmarks and MuJoCo behavior cloning show that the train-then-distill pipeline produces compact LDS predictors that match or outperform directly trained baselines.

Problem

• nonlinear dynamical systems
• linear state-space models
• system identification
• spectral distillation
• convex learning

Innovation

• Spectral Distillation
• Linear State-Space Models
• Observation Spectral Filtering
• Nonlinear Dynamical Systems
• Luenberger Complexity