arXiv:2606. 28464v1 Announce Type: new Abstract: In the optimization of neural networks, gradient dynamics are influenced by critical points that arise from the model's architecture.
By Kathl\'en Kohn, Giovanni Luca Marchetti, Farhan Shabir, Vahid Shahverdi, Weisheng Wang
The paper proposes a conjecture that composing a fixed number of distinct nonconstant polynomials with a generic high‑degree polynomial produces linearly independent polynomials, extending Newman–Slater’s theorem. The authors prove the conjecture for two polynomials and for any number when the degrees are bounded, and they show how these results explain the parameter symmetries of deep fully connected neural networks with generic polynomial activations. In particular, for architectures with layer‑specific activations of increasing degree, the conjecture’s proven cases fully characterize the parameter sets that yield the same end‑to‑end network function, and it also resolves the identifiability of shallow polynomial networks.
By Kathl\'en Kohn, Giovanni Luca Marchetti, Alex Massarenti, Massimiliano Mella
arXiv:2410. 00722v3 Announce Type: replace Abstract: We study convolutional neural networks with monomial activation functions.
By Vahid Shahverdi, Giovanni Luca Marchetti, Kathl\'en Kohn
arXiv:2605.09609v3 Announce Type: replace
Abstract: Minimal filling architectures describe the smallest hidden-layer widths for which the neurovariety of a polynomial neural network fills its ambient...
By Kevin Dao, Jose Israel Rodriguez
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:2605. 09609v2 Announce Type: replace Abstract: We provide counterexamples to the unimodal minimal filling architecture conjecture for polynomial neural networks (PNNs) with power activation functions.
By Kevin Dao, Jose Israel Rodriguez