Gaussian Universality and Its Breakdown in Tensor-Network Machine Learning
Abstract
Gaussian-process limits are powerful in describing overparameterized machine learning models, yet their validity in structured tensor-network architectures remains unclear. Here we analytically present a moment-based approach that identifies precise conditions for the emergence and breakdown of Gaussian universality in tensor-network learning models, with a focus on matrix product states. We prove that in the large bond dimension limit, the learning models with both local and global observables converge to Gaussian processes, with explicit finite-size bounds on higher-order moment deviations. Whereas in the large physical dimension limit, the Gaussian universality no longer persists: while the models with local observables retain Gaussian-process behavior, those global cases exhibit persistent non-Gaussian corrections. Our results reveal that Gaussian-process behavior in tensor-network learning is controlled not only by parameter number, but also by architectural scaling, observable locality, and the spectral properties.
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BibTeX
@misc{wang2026gaussian,
title = {{Gaussian Universality and Its Breakdown in Tensor-Network Machine Learning}},
author = {Shi-Tuan Wang and Zidu Liu and Li-Wei Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06080},
eprint = {2610.06080},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06080},
}
RIS
TY - UNPB
TI - Gaussian Universality and Its Breakdown in Tensor-Network Machine Learning
AU - Shi-Tuan Wang
AU - Zidu Liu
AU - Li-Wei Yu
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06080
UR - https://arxiv.org/abs/2610.06080
AB - Gaussian-process limits are powerful in describing overparameterized machine learning models, yet their validity in structured tensor-network architectures remains unclear. Here we analytically present a moment-based approach that identifies precise conditions for the emergence and breakdown of Gaussian universality in tensor-network learning models, with a focus on matrix product states. We prove that in the large bond dimension limit, the learning models with both local and global observables converge to Gaussian processes, with explicit finite-size bounds on higher-order moment deviations. Whereas in the large physical dimension limit, the Gaussian universality no longer persists: while the models with local observables retain Gaussian-process behavior, those global cases exhibit persistent non-Gaussian corrections. Our results reveal that Gaussian-process behavior in tensor-network learning is controlled not only by parameter number, but also by architectural scaling, observable locality, and the spectral properties.
ER -