This work investigates the problem of estimating the weight matrices of a stable time-invariant linear dynamical system from a single sequence of noisy measurements. We show that if the unknown weight matrices describing the system are in Brunovsky canonical form, we can efficiently estimate the ground truth unknown matrices of the system from a linear system of equations formulated based on the transfer function of the system, using both online and offline stochastic gradient descent (SGD) methods. Specifically, by deriving concrete complexity bounds, we show that SGD converges linearly in expectation to any arbitrary small Frobenius norm distance from the ground truth weights. To the best of our knowledge, ours is the first work to establish linear convergence characteristics for online and offline gradient-based iterative methods for weight matrix estimation in linear dynamical systems from a single trajectory. Extensive numerical tests verify that the performance of the proposed methods is consistent with our theory, and show their superior performance relative to existing state of the art methods.

中文翻译：

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在线随机梯度下降从单个轨迹学习线性动力学系统

这项工作研究了从单个噪声测量序列估计稳定的时不变线性动力学系统的权重矩阵的问题。我们表明，如果描述该系统的未知权重矩阵为Brunovsky典范形式，则可以使用基于系统传递函数的线性方程组，使用在线和离线方法有效地估计系统的地面实数未知矩阵随机梯度下降（SGD）方法。具体而言，通过推导具体的复杂度边界，我们表明SGD期望线性收敛于从地面真实权重得出的任意小Frobenius范数距离。据我们所知，我们的工作是建立基于在线和离线基于梯度的迭代方法的线性收敛特性的第一项工作，该迭代方法用于从单个轨迹对线性动力系统中的权重矩阵进行估计。大量的数值测试验证了所提出方法的性能与我们的理论是一致的，并显示出它们相对于现有技术水平的优越性能。