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On the Surprising Effectiveness of Spectral Clipping in Learning Stable Linear and Latent-Linear Dynamical Systems
Hanyao Guo, Yunhai Han, Harish Ravichandar
TL;DR
Unconstrained learning of linear dynamical systems can yield unstable predictions, while existing stability-enforcing approaches add computational burden. This paper applies post-hoc spectral clipping to learned system matrices and combines it with Koopman methods for nonlinear systems. Experiments report comparable or better predictive accuracy, stable long-horizon predictions, and orders-of-magnitude speedups; the nonlinear stability claim concerns learned open-loop predictions.
Problem
Unconstrained learning can produce unstable systems and catastrophic long-term predictions, while existing stability-enforcing methods can introduce considerable computational burden.
Source: §1 · p. 2Method
Spectral clipping applies unconstrained least squares, clips eigenvalue magnitudes above one, and reconstructs the system matrix with unchanged eigenvectors; it can also be combined with Koopman operators.
Source: §1 · p. 2Results
SC achieves comparable or better predictive accuracy than strong baselines, stable long-horizon predictions, and orders-of-magnitude faster computation; it also supports stable predictions from unsuccessful or truncated robot demonstrations.
Source: §1 · pp. 2–3Takeaways & Limitations
SC provides a post-hoc way to obtain stable learned linear systems and can be combined with Koopman methods for nonlinear systems.
Source: §1 · pp. 2–3The nonlinear-system stability claim concerns the learned open-loop predictor and does not imply closed-loop stability without further assumptions.
Source: §4 · p. 6
Abstract
from arXiv · showhide
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.
1. Introduction
Linear dynamical systems remain useful for their analytical solutions, low computational burden, and role in Koopman-based modeling, but unconstrained learning can produce unstable systems. The paper introduces spectral clipping and evaluates its accuracy, stability, efficiency, nonlinear-system use, and robustness to imperfect robot demonstrations.
1. Introduction
- Unconstrained learning can produce unstable systems and catastrophic long-term predictions, while stability-enforcing alternatives often add considerable computation.The paper frames the challenge as retaining accuracy and efficiency while ensuring stability.Source: §1 · p. 2
- SC learns a system matrix with unconstrained least squares, clips eigenvalue magnitudes above one, and reconstructs the matrix using unchanged eigenvectors.The approach is presented as a post-hoc stability procedure with little extra computation.Source: §1 · p. 2
- SC combines with Koopman operators to learn nonlinear systems with improved predictive accuracy and stability, including dexterous manipulation dynamics learned in seconds.The example involves behavioral dynamics of multi-fingered robotic-hand manipulation skills.Source: §1 · p. 2
- SC consistently achieves comparable or better predictive accuracy than CG, WLS, and SOC while producing stable long-horizon predictions and running orders of magnitude faster.Experiments span two video-prediction datasets and a robot model-learning dataset.Source: §1 · p. 2
- SC supports robust, stable predictions from robot training data containing unsuccessful or truncated demonstrations, unlike unconstrained learning, which can yield erratic, ineffective behavior.The paper notes these data characteristics are common in real-world robot-learning datasets.Source: §1 · p. 3
2. Background: Learning Stable Linear Dynamical Systems
The background formulates learning a stable linear dynamical system from trajectories and contrasts unconstrained least squares with stability-constrained learning. The former does not ensure stability, while the latter can be non-convex and require approximations.
2. Background: Learning Stable Linear Dynamical Systems
- Trajectory data for a stable autonomous LDS comprises state sequences from which paired data matrices X and Y represent consecutive states.The system evolves as x_(t+1) = A x_t, with A assumed Schur stable.Source: §2 · p. 3
- Unconstrained least squares fits the system matrix to the data but imposes no stability constraint, so the learned system may be unstable despite a stable source system.The objective minimizes a Frobenius-norm prediction error.Source: §2 · p. 3
- Stability-constrained learning explicitly restricts the optimization to stable matrices, but the resulting problem is non-convex and requires approximations.The constrained solution is guaranteed stable.Source: §2 · p. 3
3. Post-Hoc Stability via Spectral Clipping
Spectral clipping is a post-hoc procedure that first estimates a system matrix by least squares, then clips unstable eigenvalues and reconstructs the matrix with its eigenvectors unchanged. The clipping margin controls stability and expressivity, while the method is designed to retain predictive accuracy.
3. Post-Hoc Stability via Spectral Clipping
- SC first estimates the system matrix by unconstrained least squares, decomposes it into eigenvalues and eigenvectors, clips unstable eigenvalues, and reconstructs the matrix.The reconstruction uses the clipped eigenvalues and the original modal matrix.Source: §3 · p. 4
- Clipping eigenvalues above magnitude one and retaining the eigenvectors changes modal rates without changing the system’s invariant subspace.The geometric intuition is that eigenvalues govern whether trajectories grow, decay, or oscillate.Source: §3 · pp. 4–5, Figure 1 · p. 4
- For non-diagonalizable estimates, tiny perturbations can yield a nearby diagonalizable matrix to which SC can be applied with arbitrarily small approximation error.The paper states that the resulting system evolution can be made arbitrarily close to that of the original matrix.Source: §3 · p. 4
- The paper extends SC beyond stable LDS learning to nonlinear systems and systems with control inputs.For nonlinear systems, it uses Koopman-based methods; control-input details are in the appendix.Source: §3 · p. 5