We study the problem of identification of linear dynamical system from a single trajectory, via excitations of isotropic Gaussian. In stark contrast with previously reported results, Ordinary Least Squares (OLS) estimator for even \emph{stable} dynamical system contains non-vanishing error in \emph{high dimensions}; which stems from the fact that realizations of non-diagonalizable dynamics can have strong \emph{spatial correlations} and a variance, of order $O(e^{n})$, where $n$ is the dimension of the underlying state space. Employing \emph{concentration of measure phenomenon}, in particular tensorization of \emph{Talagrands inequality} for random dynamical systems we show that observed trajectory of dynamical system of length-$N$ can have a variance of order $O(e^{nN})$. Consequently, showing some or most of the $n$ distances between an $N-$ dimensional random vector and an $(n-1)$ dimensional hyperplane in $\mathbb{R}^{N}$ can be close to zero with positive probability and these estimates become stronger in high dimensions and more iterations via \emph{Isoperimetry}. \emph{Negative second moment identity}, along with distance estimates give a control on all the singular values of \emph{Random matrix} of data, revealing limitations of OLS for stable non-diagonalizable and explosive diagonalizable systems.

翻译：我们研究通过各向同性高斯激励从单条轨迹辨识线性动力系统的问题。与已有结果截然不同的是，即使对\emph{稳定}动力系统，普通最小二乘（OLS）估计量在\emph{高维}情形下仍存在非零误差；这一现象源于不可对角化动力学的实现可能具有强\emph{空间相关性}，且方差量级为$O(e^{n})$，其中$n$为底层状态空间的维度。利用\emph{测度集中现象}，特别是随机动力系统的\emph{Talagrand不等式}的张量化，我们证明长度为$N$的动力系统观测轨迹的方差量级可达$O(e^{nN})$。进而表明：在$\mathbb{R}^{N}$中，$N$维随机向量与$(n-1)$维超平面之间的$n$个距离中，部分或大部分以正概率接近于零，且这些估计通过\emph{等周原理}在高维和多次迭代下更强。结合距离估计的\emph{负二阶矩恒等式}可控制数据\emph{随机矩阵}的所有奇异值，揭示了OLS对稳定不可对角化系统及爆炸可对角化系统的局限性。