High dimensional random dynamical systems are ubiquitous, including -- but not limited to -- cyber-physical systems, daily return on different stocks of S&P 1500 and velocity profile of interacting particle systems around McKeanVlasov limit. Mathematically, underlying phenomenon can be captured via a stable $n$-dimensional linear transformation `$A$' and additive randomness. System identification aims at extracting useful information about underlying dynamical system, given a length $N$ trajectory from it (corresponds to an $n \times N$ dimensional data matrix). We use spectral theorem for non-Hermitian operators to show that spatio-temperal correlations are dictated by the discrepancy between algebraic and geometric multiplicity of distinct eigenvalues corresponding to state transition matrix. Small discrepancies imply that original trajectory essentially comprises of multiple lower dimensional random dynamical systems living on $A$ invariant subspaces and are statistically independent of each other. In the process, we provide first quantitative handle on decay rate of finite powers of state transition matrix $\|A^{k}\|$ . It is shown that when a stable dynamical system has only one distinct eigenvalue and discrepancy of $n-1$: $\|A\|$ has a dependence on $n$, resulting dynamics are spatially inseparable and consequently there exist at least one row with covariates of typical size $\Theta\big(\sqrt{N-n+1}$ $e^{n}\big)$ i.e., even under stability assumption, covariates can suffer from curse of dimensionality. In the light of these findings we set the stage for non-asymptotic error analysis in estimation of state transition matrix $A$ via least squares regression on observed trajectory by showing that element-wise error is essentially a variant of well-know Littlewood-Offord problem.

翻译：高维随机动力系统普遍存在，包括但不限于信息物理系统、标普1500指数中不同股票的日收益率，以及McKean-Vlasov极限附近相互作用粒子系统的速度分布。从数学角度看，潜在的动态现象可通过稳定的n维线性变换"A"和加性随机性来刻画。系统辨识旨在从给定长度为N的轨迹（对应n×N维数据矩阵）中提取关于底层动力系统的有用信息。我们利用非厄米算子的谱定理证明，时空相关性由状态转移矩阵对应不同特征值的代数量数与几何量数之差决定。当差值较小时，原始轨迹本质上由多个存在于A不变子空间上的低维随机动力系统组成，且这些系统彼此统计独立。在此过程中，我们首次给出了状态转移矩阵有限幂次衰减速率的定量刻画。研究表明，当稳定动力系统仅有一个不同特征值且差距为n-1时：‖A‖依赖于n，导致动力学在空间上不可分离，进而至少存在一行协变量具有典型量级Θ(√(N-n+1) e^n) ——即使在稳定性假设下，协变量仍可能遭遇维数灾难。基于这些发现，我们通过证明元素级误差本质上是著名Littlewood-Offord问题的一个变体，为通过观测轨迹上的最小二乘回归估计状态转移矩阵A的非渐近误差分析奠定基础。