论文标题
提高了预测和鉴定部分观察到的线性动力学系统的速率
Improved rates for prediction and identification of partially observed linear dynamical systems
论文作者
论文摘要
从部分观察中识别线性时间不变的动力系统是控制理论中的一个基本问题。特别具有挑战性的是展示长期记忆的系统。一个自然的问题是，如何根据固有的维度（顺序）$ d $而不是更大的内存长度来学习非质子统计率的学习系统。我们提出了一种算法，该算法在高斯观察噪声中给出了一个长度$ t $的单个轨迹，以几乎最佳的速率$ \ widetilde o \ left（\ sqrt \ sqrt \ frac {d} {t} {t} \ right）$ in $ \ \ \ \ \ \ mathcal {我们还为过程噪声提供了界限，并改善了学习系统实现的界限。我们的算法基于多尺度的低级别近似：SVD应用于几何增加尺寸的Hankel矩阵。我们的分析依赖于在傅立叶域上仔细应用浓度界限 - 我们为相关输入的样本协方差和$ \ Mathcal H_ \ Infty $ norm估计提供了更清晰的浓度界限，这可能是独立的。
Identification of a linear time-invariant dynamical system from partial observations is a fundamental problem in control theory. Particularly challenging are systems exhibiting long-term memory. A natural question is how learn such systems with non-asymptotic statistical rates depending on the inherent dimensionality (order) $d$ of the system, rather than on the possibly much larger memory length. We propose an algorithm that given a single trajectory of length $T$ with gaussian observation noise, learns the system with a near-optimal rate of $\widetilde O\left(\sqrt\frac{d}{T}\right)$ in $\mathcal{H}_2$ error, with only logarithmic, rather than polynomial dependence on memory length. We also give bounds under process noise and improved bounds for learning a realization of the system. Our algorithm is based on multi-scale low-rank approximation: SVD applied to Hankel matrices of geometrically increasing sizes. Our analysis relies on careful application of concentration bounds on the Fourier domain -- we give sharper concentration bounds for sample covariance of correlated inputs and for $\mathcal H_\infty$ norm estimation, which may be of independent interest.
