具有非线性相关元素的矩阵的谱普适性
Spectral Universality for Matrices with Non-Linear Correlated Entries
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中文总结 AI 辅助
本文针对元素为独立同分布随机场非线性函数的矩阵，在指数衰减条件下建立了其谱与高斯矩阵及自由模型的非渐近谱比较，并给出了非交换Khintchine型界，适用于因果线性过程、Volterra过程和神经网络等场景。
中文摘要 AI 辅助
我们针对元素为独立同分布随机场的非线性函数的矩阵，建立了非渐近谱比较结果。在$\mathbb L^\infty$-耦合系数的指数衰减假设下，我们推导了这些矩阵的谱与具有匹配协方差结构的高斯矩阵的谱之间的豪斯多夫距离的高概率界。我们进一步推导了与协方差匹配的自由模型的比较，并得到了相应的非交换Khintchine型界。这些结果尤其适用于元素为因果线性过程、Volterra型过程和神经网络的非线性变换的矩阵。证明结合了有限记忆近似、块分解、高斯插值和预解估计。
英文摘要
We establish nonasymptotic spectral comparison results for matrices whose entries are non-linear functions of an iid random field. Under an exponential decay assumption on an $\mathbb L^\infty$-coupling coefficient, we derive high-probability bounds for the Hausdorff distance between their spectra and those of Gaussian matrices with matching covariance structures. We further derive a comparison with a covariance-matched free model and derive corresponding noncommutative Khintchine-type bounds. The results apply, in particular, to matrices whose entries are nonlinear transformations of causal linear processes, Volterra-type processes, and neural networks. The proof combines finite-memory approximation, block decomposition, Gaussian interpolation, and resolvent estimates.
