论文标题
从单个轨迹学习非线性动力系统
Learning nonlinear dynamical systems from a single trajectory
论文作者
论文摘要
我们介绍了用于学习$ x_ {t+1} =σ（θ^{\ star} x_t）+\ varepsilon_t $的非线性动力学系统的算法，其中$θ^{\ star} $是一个非线性链接函数，$ ques nonlinear link function unight unight ins unight unight unight unight $ \ vareps $ \ vareps $ a。我们给出了一种算法，该算法从具有最佳样本复杂性和线性运行时间的单个轨迹中恢复了重量矩阵$θ^{\ star} $。该算法在较弱的统计假设下取得了成功，尤其是i）不需要绑定重量矩阵$θ^{\ star} $的光谱规范（而不是光谱半径的概括）和ii）享受非局限性链接的保证，例如crialu surnu function in cor cor cor cor unu functue。我们的分析有两个关键组成部分：i）我们提供了一个通用配方，从而可以使用全球稳定性的非线性动力学系统稳定性来证明状态矢量协方差是否有良好的条件，ii）使用这些工具，我们扩展了众所周知的算法，以有效地学习通用线性模型。
We introduce algorithms for learning nonlinear dynamical systems of the form $x_{t+1}=σ(Θ^{\star}x_t)+\varepsilon_t$, where $Θ^{\star}$ is a weight matrix, $σ$ is a nonlinear link function, and $\varepsilon_t$ is a mean-zero noise process. We give an algorithm that recovers the weight matrix $Θ^{\star}$ from a single trajectory with optimal sample complexity and linear running time. The algorithm succeeds under weaker statistical assumptions than in previous work, and in particular i) does not require a bound on the spectral norm of the weight matrix $Θ^{\star}$ (rather, it depends on a generalization of the spectral radius) and ii) enjoys guarantees for non-strictly-increasing link functions such as the ReLU. Our analysis has two key components: i) we give a general recipe whereby global stability for nonlinear dynamical systems can be used to certify that the state-vector covariance is well-conditioned, and ii) using these tools, we extend well-known algorithms for efficiently learning generalized linear models to the dependent setting.
