论文标题
部分可观测时空混沌系统的无模型预测
Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity
论文作者
论文摘要
由于当前对理解高维空间中随机景观的统计特性的兴趣的兴趣，我们考虑了$ \ Mathbb {r}^n $中的景观模型，该模型是通过叠加$ m> n $平面随机波形传播和振幅所获得的。对于此景观，我们展示了如何计算“退火复杂性”，以控制平均固定点的渐近增长率为$ n \ to \ infty $，以固定比率$α= m/n> 1 $。该计算的框架要求我们研究$ n \ times n $矩阵$ w = ktk^t $的光谱属性，其中$ t $是对角线的，$ m $平均零i.i.d.真实的正态分布条目，以及$ k $的所有$ MN $条目也是I.I.D.实际正常随机变量。我们建议将后者称为高斯Marchenko-Pastur合奏，因为这些矩阵出现在1967年的开创性论文中。我们计算相关的平均光谱密度，并评估涉及特征多项式和相关矩阵的特征多项式产物的某些矩和相关函数。
Motivated by current interest in understanding statistical properties of random landscapes in high-dimensional spaces, we consider a model of the landscape in $\mathbb{R}^N$ obtained by superimposing $M>N$ plane waves of random wavevectors and amplitudes. For this landscape we show how to compute the "annealed complexity" controlling the asymptotic growth rate of the mean number of stationary points as $N\to \infty$ at fixed ratio $α=M/N>1$. The framework of this computation requires us to study spectral properties of $N\times N$ matrices $W=KTK^T$, where $T$ is diagonal with $M$ mean zero i.i.d. real normally distributed entries, and all $MN$ entries of $K$ are also i.i.d. real normal random variables. We suggest to call the latter Gaussian Marchenko-Pastur Ensemble, as such matrices appeared in the seminal 1967 paper by those authors. We compute the associated mean spectral density and evaluate some moments and correlation functions involving products of characteristic polynomials for such and related matrices.
