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.

Quantum Physics (quant-ph); Machine Learning (cs.LG)

arXiv:2610.06080 [quant-ph]

(or arXiv:2610.06080v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.06080 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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