The paper studies how grouping Layer Normalization (LN) units influences the approximation capacity of neural networks as their width increases. It shows that a shallow LN-Net with a single LN unit cannot approximate arbitrary functions regardless of width, but Parallel Layer Normalization networks (PLN-Nets) become universal as the number of parallel groups increases. The authors extend this result to deep networks, RMSNorm variants, feed‑forward and convolutional architectures, and validate the practical benefits of PLN through controlled MLP and physics‑informed neural network experiments, while also demonstrating that group count, grouped axes, replacement site, and label structure jointly affect fitting and generalization in CNN and ViT settings.
By Yunhao Ni, Yuxin Guo, Yuhe Liu, Wenxin Sun, Jie Luo, Wenjun Wu, Lei Huang
arXiv:2604. 20219v2 Announce Type: replace Abstract: Depth is widely viewed as a central contributor to the success of deep neural networks, whereas standard neural network approximation theory typically provides guarantees only for the final output and leaves the role of intermediate layers largely unclear.
By Shijun Zhang, Zuowei Shen, Yuesheng Xu
The paper introduces Shallow Universal Polynomial Networks (SUPNs), a single‑hidden‑layer architecture that replaces standard activations with learnable multivariate polynomials, thereby increasing expressivity per unit. SUPNs achieve universal approximation by either widening the network or increasing polynomial complexity, and the authors prove convergence rates and quasi‑optimal parameter choices. Empirical tests across a wide range of smooth, nonsmooth, and discontinuous functions in up to ten dimensions show that SUPNs often outperform deep neural networks and Kolmogorov‑Arnold networks in accuracy and variability while requiring fewer parameters and lower training cost.
By Zachary Morrow, Michael Penwarden, Brian Chen, Aurya Javeed, Akil Narayan, John D. Jakeman
arXiv:2403.04545v4 Announce Type: replace
Abstract: Scaling factors in residual branches have emerged as a prevalent method for boosting neural network performance, especially in normalization-free a...
By Zixiong Yu, Guhan Chen, Jianfa Lai, Bohan Li, Songtao Tian
The paper introduces Sliding Networks, a family of residual neural network structures that aim to preserve both feature learning and feature diversity while scaling width and depth. By maintaining a limiting ratio of residual width to the product of hidden width and depth, the authors derive a joint scaling recipe that converges to a stochastic differential equation (SDE) limit with locally maximal feature learning. Experiments on residual-MLP and language models demonstrate that this scaling approach allows redistribution of computational resources between width and depth, though performance varies when parameters are regrouped at a fixed count.
By Datong Zhou, Blake Bordelon
Transferable Graph Metanetworks extend the graph metanetwork paradigm to predict properties of neural networks of varying widths. By enforcing invariance to width-dependent representations and continuity across similar functions, the proposed modifications enable robust size generalization, especially for networks trained under the maximal‑update parameterization (μP) up to 42× the training width. The authors provide empirical evidence across multiple tasks and theoretical guarantees in the infinite‑width limit, explaining both successes and limitations under different parameterizations.
By Yuxin Ma, Adir Dayan, Yam Eitan, Haggai Maron, Soledad Villar
The paper presents a near-complete, nonasymptotic generalization theory for multilayer neural networks using path regularization, applicable to broad Lipschitz loss functions without requiring bounded loss or extreme network hyperparameters. It provides an explicit upper bound that addresses approximation rates in generalized Barron spaces and demonstrates the double descent phenomenon for ReLU networks. The authors claim near-minimax optimality for regression problems and plan to establish matching lower bounds in future work.
By Hao Yu
arXiv:2610. 25116v2 Announce Type: replace Abstract: We study positive, bias-free, one-hidden-layer ReLU networks under standard Gaussian inputs.
By Zhijie Zheng
arXiv:2607. 04597v1 Announce Type: new Abstract: In this paper, we study the universal approximation property of residual neural networks, and obtain some new results.
By Qi Zhou, Xuan Zhou, Xiao-Song Yang
arXiv:2606. 26705v1 Announce Type: cross Abstract: Feedforward neural network (NN) expressivity is typically studied by emulating optimal basis-expansion schemes.
By Anastasis Kratsios, Simone Brugiapaglia, Bum Jun Kim, Gregory Cousins, Haitz S\'aez de Oc\'ariz Borde
arXiv:2505. 22578v2 Announce Type: replace Abstract: The optimization of neural networks under weight decay remains poorly understood from a theoretical standpoint.
By Etienne Boursier, Matthew Bowditch, Matthias Englert, Ranko Lazic
arXiv:2610.30729v2 Announce Type: replace-cross
Abstract: Recent work established that the infinite-width and infinite-depth limits of residual networks commute at initialization when residual branch...
By Ruijia Zhang, Soufiane Hayou