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.

翻译：从数据中学习稳定线性动力系统时，三个重要属性是期望的：i) 预测准确性，ii) 可验证的稳定性，以及 iii) 计算效率。无约束地最小化预测误差可实现高准确性和效率，但无法保证稳定性。现有强制稳定性的方法通常能保持准确性，但往往以增加计算量为代价。在本工作中，我们研究一种看似简单的方法是否能同时满足所有三个期望。具体而言，我们考虑一种事后处理程序，即在通过无约束最小二乘法学习线性系统后，对其谱进行精细操作。我们称此方法为谱截断（SC），因为它涉及特征分解，并在将任何模超过1的特征值截断至1（不改变特征向量）后重构系统矩阵。我们还表明，SC可轻松与Koopman算子结合，以学习能够生成非线性现象稳定预测的非线性动力系统，例如涉及多指机器人手的复杂灵巧操作技能所基于的现象。通过涉及两个不同应用和公开可用基准数据集的综合实验，我们证明这种简单技术能够高效学习高度准确且可证明稳定的预测动力学。值得注意的是，我们发现SC能够匹配或超越强基线方法，同时计算速度快数个数量级。最后，我们发现即使训练数据包含失败或截断的演示，SC也能学习稳定的机器人策略。我们的代码和数据集可在 https://github.com/GT-STAR-Lab/spec_clip 找到。