We introduce algorithms for learning nonlinear dynamical systems of the form $x_{t+1}=\sigma(\Theta^{\star}x_t)+\varepsilon_t$, where $\Theta^{\star}$ is a weight matrix, $\sigma$ is a nonlinear link function, and $\varepsilon_t$ is a mean-zero noise process. We give an algorithm that recovers the weight matrix $\Theta^{\star}$ 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 $\Theta^{\star}$ (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.

该研究介绍了一种算法用于从单个轨迹中恢复权重矩阵，并成功应用于非线性动力系统，无需对其进行谱范数的限制。该算法的计算复杂度为线性，并具有较低的采样复杂度。

从单个轨迹中学习非线性动态系统