Stationary covariance spectra of discrete-time non-normal random recurrent dynamics
Abstract
Principal component analysis is widely used to characterize structure in the dynamics of recurrent neural networks. For stationary noise-driven dynamics, the distribution of variance among the principal components is determined by the spectrum of the stationary covariance matrix. While the spectral properties of this matrix are well-understood for linear networks with normal synaptic weight matrices, our understanding of the stationary covariance spectrum for random non-normal dynamics remains incomplete. In this note, we use a free-probability approach to formally derive a closed functional equation for the moment generating series of the limiting stationary covariance spectrum of discrete-time dynamics with random non-normal Gaussian weights. This characterization allows us to analyze the behavior of tail eigenvalues in the critical regime: they decay with the inverse-square of their rank. In contrast, applying the same approach to the analogous continuous-time dynamics leads to an infinite hierarchy of Schwinger-Dyson equations, rather than a closed scalar equation. We conclude with some comments regarding the relevance of these results to comparisons of models of non-normal dynamics to neural data.
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BibTeX
@misc{zavatoneveth2026stationary,
title = {{Stationary covariance spectra of discrete-time non-normal random recurrent dynamics}},
author = {Jacob A. Zavatone-Veth},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.31944},
eprint = {2606.31944},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.31944},
}
RIS
TY - UNPB
TI - Stationary covariance spectra of discrete-time non-normal random recurrent dynamics
AU - Jacob A. Zavatone-Veth
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2606.31944
UR - https://arxiv.org/abs/2606.31944
AB - Principal component analysis is widely used to characterize structure in the dynamics of recurrent neural networks. For stationary noise-driven dynamics, the distribution of variance among the principal components is determined by the spectrum of the stationary covariance matrix. While the spectral properties of this matrix are well-understood for linear networks with normal synaptic weight matrices, our understanding of the stationary covariance spectrum for random non-normal dynamics remains incomplete. In this note, we use a free-probability approach to formally derive a closed functional equation for the moment generating series of the limiting stationary covariance spectrum of discrete-time dynamics with random non-normal Gaussian weights. This characterization allows us to analyze the behavior of tail eigenvalues in the critical regime: they decay with the inverse-square of their rank. In contrast, applying the same approach to the analogous continuous-time dynamics leads to an infinite hierarchy of Schwinger-Dyson equations, rather than a closed scalar equation. We conclude with some comments regarding the relevance of these results to comparisons of models of non-normal dynamics to neural data.
ER -