arXiv:2511. 19703v2 Announce Type: replace-cross Abstract: We study the dimension and identifiability of neurovarieties associated to polynomial neural networks.
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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.
arXiv:2508. 03867v2 Announce Type: replace-cross Abstract: We introduce a class of algebraic varieties naturally associated with ReLU neural networks, arising from the piecewise linear structure of their outputs across activation regions in input space, and the piecewise multilinear structure in parameter space.
arXiv:2410. 00722v3 Announce Type: replace Abstract: We study convolutional neural networks with monomial activation functions.
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.
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.
arXiv:2609. 28500v1 Announce Type: cross Abstract: We study exact representability by bias-free shallow polynomial neural networks using algebraic geometry.