从有限轨迹学习遍历动力系统
Learning Ergodic Dynamical Systems from a Finite Trajectory
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中文总结 AI 辅助
研究从遍历随机动力系统的有限轨迹学习，用非线性最小二乘法估计最优单步预测函数并得高概率保证，扩展框架到高阶系统和有限状态空间，还表明该方法可用于学习库普曼算子。
中文摘要 AI 辅助
我们考虑从遍历随机动力系统的单个有限轨迹进行学习的问题。具体而言，研究定义时间齐次马尔可夫过程的离散时间自治随机系统。首先通过非线性最小二乘法估计最优单步预测函数，并得出关于过程不变测度的高概率保证。接着将框架扩展到高阶系统和有限状态空间。最后表明相同的最小二乘法和集中论证自然地扩展到学习库普曼算子。我们的方法结合了统计学习理论和马尔可夫链的定量遍历理论的工具。
英文摘要
We consider the problem of learning from a single finite trajectory of an ergodic stochastic dynamical system. More precisely, we study discrete-time autonomous stochastic systems defining time-homogeneous Markov processes. We first focus on estimating the optimal one-step prediction function by nonlinear least squares, and derive high-probability guarantees measured with respect to the invariant measure of the process. These results make explicit how the non-independent and non-identically distributed nature of trajectory data modifies the classical statistical learning analysis. We then extend the framework to higher-order systems and finite-state spaces. Finally, we show that the same least squares and concentration arguments naturally extend to learning Koopman operators. Our approach combines tools from statistical learning theory and quantitative ergodic theory for Markov chains. It relies, in particular, on a concentration inequality for Hilbert-space-valued additive functionals of uniformly geometrically ergodic Markov chains.
发表机构
- MaLGa center, DIMA, Università degli Studi di Genova（马尔加中心、DIMA、热那亚大学）
- Istituto Italiano di Tecnologia, Genoa（意大利技术研究院、热那亚）
机构由 AI 辅助整理，请以论文原文为准。
