The paper proves the weak composition‑independence conjectures of Kohn, Marchetti, Massarenti and Mella by employing deformations of rational curves and a single kernel of the linearized equation, with Riemann‑Roch providing a contradiction from tangent fields with controlled poles. This result establishes that a neural network using a general polynomial activation function of sufficiently large degree is identifiable.
By Xiaotian Chang, Yacoub Hendi
arXiv:2511. 19703v2 Announce Type: replace-cross Abstract: We study the dimension and identifiability of neurovarieties associated to polynomial neural networks.
By A. Massarenti, M. Mella
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:2609.05572v1 Announce Type: new
Abstract: We prove that every strictly positive probability distribution on \(\{-1,1\}^n\) is represented exactly by a sigmoid belief network with finite paramet...
By Gleb Smirnov
arXiv:2607. 07778v1 Announce Type: new Abstract: Bubeck, Li and Nagaraj conjectured that, for generic data, any two-layer neural network with $m$ neurons that fits $n$ noisy labels must have Lipschitz constant at least of order $\sqrt{n/m}$, with no restriction on the size of the weights.
By Yitzchak Shmalo
arXiv:2606. 16579v1 Announce Type: new Abstract: We extend the entropy formula of Menon and Yu for the real Deep Linear Network (DLN) to its complex and quaternionic analogues, obtaining a unified formula for DLNs over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$.
By Luis Contreras, Marco Nahas, Tejas Kotwal
arXiv:2606. 10806v1 Announce Type: new Abstract: Moonshine is an autonomous agent whose central objective is to generate mathematical conjectures.
By Xiaoyang Chen, Xiang Jiang
The paper investigates the limits of implementing neural network field theory on a computer, focusing on function classes that are regular enough for computation. It presents a no‑go theorem showing that finite‑width network ensembles cannot consistently realize either a quantum or effective field theory due to violations of reflection positivity and lack of scale separation. The study distinguishes between finite‑width and infinite‑width interpretations, concluding that only smeared correlators of the infinite‑width limit are computable with controlled error, and identifies two possible ways to evade the theorem—by relaxing finite variance or exact rotation invariance.
By Thomas R. Harvey
arXiv:2606. 09820v1 Announce Type: cross Abstract: We generalize the universal approximation theorem for functional input neural networks (FNN) to differentiable maps by including the approximation of the derivatives.
By Philipp Schmocker, Josef Teichmann
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:2606. 16975v1 Announce Type: cross Abstract: In this work, we investigate new activation functions for achieving arbitrary-accuracy Sobolev approximation by fixed-size neural networks.
By Baicheng Li, Haizhao Yang, Shijun Zhang
The paper investigates topological expressivity of neural networks by examining Betti numbers of the network output rather than decision boundaries. It proves that a ReLU network with a specific architecture can represent spaces homotopy equivalent to wedge sums of spheres with arbitrary Betti numbers, and provides tighter bounds for spaces equivalent to products of spheres. The authors also conjecture an even stronger bound for networks representing a single sphere.
By Kosio Beshkov