Parallel Layer Normalization for Universal Approximation

Abstract:Layer Normalization (LN) is usually studied for its role in optimization, but it also acts as a multivariate nonlinearity between affine layers. We investigate how grouping LN units affects the approximation power of neural networks as width grows. In Parallel Layer Normalization networks (PLN-Nets), fixed-dimensional LN units are applied independently to groups between two affine layers. We prove that a shallow LN-Net with one LN unit is non-universal regardless of width, whereas PLN-Nets become universal as the number of parallel groups grows. The mechanism extends to deep networks, Root Mean Square Normalization (RMSNorm) variants, position-wise feed-forward networks, and convolutional networks under suitable grouping structures. Controlled MLP and physics-informed neural network (PINN) experiments support PLN's practical viability, while CNN and ViT experiments show that group count shapes the finite-sample fitting envelope but does not act alone: the grouped axes, replacement site, and label structure lead to distinct fitting and generalization behavior.

34 pages, 11 figures. Revised theoretical analysis, proofs, and experiments

Machine Learning (cs.LG); Machine Learning (stat.ML)

arXiv:2505.13142 [cs.LG]

(or arXiv:2505.13142v3 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2505.13142 Focus to learn more arXiv-issued DOI via DataCite

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