On the Surprising Effectiveness of Spectral Clipping in Learning Stable Linear and Latent-Linear Dynamical Systems

When learning stable linear dynamical systems from data, three important properties are desirable: i) predictive accuracy, ii) verifiable stability, and iii) computational efficiency. Unconstrained minimization of prediction errors leads to high accuracy and efficiency but cannot guarantee stability. Existing methods to enforce stability often preserve accuracy, but do so only at the cost of increased computation. In this work, we investigate if a seemingly-naive procedure can simultaneously offer all three desiderata. Specifically, we consider a post-hoc procedure in which we surgically manipulate the spectrum of the linear system after it was learned using unconstrained least squares. We call this approach spectral clipping (SC) as it involves eigen decomposition and subsequent reconstruction of the system matrix after any eigenvalues whose magnitude exceeds one have been clipped to one (without altering the eigenvectors). We also show that SC can be readily combined with Koopman operators to learn nonlinear dynamical systems that can generate stable predictions of nonlinear phenomena, such as those underlying complex dexterous manipulation skills involving multi-fingered robotic hands. Through comprehensive experiments involving two different applications and publicly available benchmark datasets, we show that this simple technique can efficiently learn highly-accurate predictive dynamics that are provably-stable. Notably, we find that SC can match or outperform strong baselines while being orders-of-magnitude faster. Finally, we find that SC can learn stable robot policies even when the training data includes unsuccessful or truncated demonstrations. Our code and datasets can be found at https://github.com/GT-STAR-Lab/spec_clip.

💡 Research Summary

The paper tackles a fundamental problem in system identification: learning discrete‑time linear dynamical systems (LDS) that are both accurate in prediction and provably stable, while keeping computational costs low. Conventional unconstrained least‑squares (LS) yields the best short‑term prediction error but can produce matrices whose spectral radius exceeds one, leading to divergent long‑horizon forecasts. Existing stability‑enforcing approaches (constraint generation, weighted LS, SOC‑based characterizations) embed stability constraints directly into the optimization, which guarantees a stable solution but at the price of heavy iterative solvers, semidefinite programming, or costly eigenvalue checks.

The authors propose a remarkably simple post‑hoc technique called Spectral Clipping (SC). The procedure is: (1) compute the LS estimate (\hat A_{LS}); (2) perform an eigen‑decomposition (\hat A_{LS}=M\Lambda M^{-1}); (3) replace any eigenvalue (\lambda_i) with magnitude (|\lambda_i|\ge 1) by (\lambda_i/(|\lambda_i|)(1-\varepsilon)) (or simply by 1 when (\varepsilon=0)); (4) reconstruct (\hat A_{SC}=M\bar\Lambda M^{-1}). Because only the eigenvalues are altered, the eigenvectors – and thus the modal directions of the system – remain unchanged. The resulting matrix is Schur‑stable by construction: its spectral radius is bounded by (1-\varepsilon) (strictly stable for (\varepsilon>0), marginally stable for (\varepsilon=0)).

The paper provides both intuition and theoretical justification. Geometrically, clipping reduces the growth rate of those modes that would otherwise explode, without rotating the state‑space axes. Analytically, the authors show that for non‑diagonalizable (\hat A_{LS}) a tiny perturbation yields a nearby diagonalizable matrix; the dynamics of the perturbed system can be made arbitrarily close to the original, so the clipping step introduces negligible approximation error.

A key contribution is the extension of SC to nonlinear dynamics via Koopman operator theory. By lifting the original state (\xi_k) into a high‑dimensional latent space (z_k=\phi(\xi_k)) (state‑inclusive lifting), a linear Koopman matrix (\hat K) is learned. Applying the same eigenvalue clipping to (\hat K) yields a stable Koopman matrix (\hat K_{SC}). Because the original state is part of the latent vector, the boundedness and decay properties of the latent predictor directly translate to the decoded physical states, guaranteeing open‑loop stability of the learned predictor (though not closed‑loop stability of the true nonlinear system).

Empirical evaluation spans three domains: two video‑prediction benchmarks (e.g., Moving MNIST, KTH) and a robotic manipulation dataset collected in simulation. Baselines include vanilla LS, Constraint Generation (CG), Weighted LS (WLS), and the SOC method. Metrics cover short‑term prediction error (RMSE, PSNR), long‑term stability (whether trajectories diverge), and wall‑clock training time. Results show that SC matches or slightly outperforms constrained baselines in accuracy while being 10–100× faster. In the robotics experiment, training data contain both successful and failed demonstrations; LS produces unstable policies that cause the simulated hand to “blow up,” whereas SC yields a stable policy that successfully reproduces the desired manipulation even when trained on truncated trajectories.

The authors also study the effect of the clipping margin (\varepsilon). Larger (\varepsilon) enforces stricter decay, leading to faster convergence but reduced expressive power (the model may under‑fit fast dynamics). Setting (\varepsilon=0) provides marginal stability and retains maximal expressivity, which the experiments indicate is sufficient for most benchmark tasks.

In summary, Spectral Clipping offers a lightweight, theoretically sound, and empirically validated route to stable LDS learning. It sidesteps the need for complex constrained optimization, scales to high‑dimensional problems (including image‑level prediction), and integrates seamlessly with Koopman‑based nonlinear system identification. The authors release code and datasets, facilitating reproducibility and encouraging adoption in control, robotics, and data‑driven dynamical modeling where stability guarantees are essential but computational resources are limited.