arXiv Machine Learning
Jun 15
On the Geometry and Optimization of Polynomial Convolutional Networks
arXiv:2410. 00722v3 Announce Type: replace Abstract: We study convolutional neural networks with monomial activation functions.
The Geometry of Polynomial Group Convolutional Neural Networks
The paper introduces a new mathematical framework for polynomial group convolutional neural networks (PGCNNs) using graded group algebras. It presents two natural parametrizations of the architecture—based on Hadamard and Kronecker products—that are related by a linear map. The authors compute the dimension of the resulting neuromanifold, show it depends only on the number of layers and group size, and describe the general fiber of the Kronecker parametrization, conjecturing a similar description for the Hadamard case, supported by explicit computations for small groups and shallow networks.
Related stories
arXiv Machine Learning
Jun 3
Theoretical Aspects of Lie Groupoid and Lie Algebroid Equivariant Convolutional Neural Networks
arXiv:2606. 02758v1 Announce Type: cross Abstract: We introduce Lie groupoid equivariant neural networks as a specialization of recently proposed topological category-equivariant neural networks to the differentiable setting.
arXiv Machine Learning
Sep 25
Algebraic Expressivity Certificates for Shallow Polynomial Neural Networks
arXiv:2609. 28500v1 Announce Type: cross Abstract: We study exact representability by bias-free shallow polynomial neural networks using algebraic geometry.
arXiv Machine Learning
Jun 16
Constraining the outputs of ReLU neural 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 Machine Learning
Sep 23
Graded Representation Theory of Equivariant Neural Networks
arXiv:2609.25776v1 Announce Type: cross Abstract: Nonlinear activations can create equivariant interactions between irreducible representations that linear maps cannot. We use the Gaussian degree dec...
arXiv Machine Learning
Jun 26
A General Framework for Learning Algebraic Properties from Cayley Graphs using Graph Neural Networks
arXiv:2606. 26212v1 Announce Type: new Abstract: A Graph Neural Network (GNN) framework for predicting the solvability of finite groups from their Cayley graph representations was introduced in [1].
arXiv Machine Learning
Jun 9
Graph Neural Networks for Predicting Solvability of Finite Groups
arXiv:2606. 07619v1 Announce Type: new Abstract: We present a Graph Neural Network (GNN) framework for the classification of finite groups according to their solvability.
arXiv Machine Learning
Jul 7
A Complete Symmetry Classification of Shallow ReLU Networks
arXiv:2604. 14037v2 Announce Type: replace Abstract: Parameter space is not function space for neural network architectures.
arXiv Machine Learning
Aug 28
Linear Independence of Polynomial Compositions and Identifiability of Deep Neural Networks
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 Machine Learning
2d ago
Parameter symmetries determine representational geometry in overparameterized nonlinear networks
arXiv:2609.39078v1 Announce Type: new Abstract: Representations are routinely used across machine learning, psychology, and neuroscience to draw inferences about the computations of biological and ar...
arXiv Machine Learning
Jun 19
Minimal Filling Architectures of Polynomial Neural Networks: Counterexamples, Frontier Search, and Defects
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 Machine Learning
Aug 27
M-Fibration Theory with Applications to Neural Network Compression
The paper introduces a general theoretical framework for fibrations on graphs labeled by a commutative monoid, extending the classic theory of graph fibrations to weighted and algebraically labeled graphs. It also accommodates approximate fibrations and demonstrates how this framework can be used to compress arbitrary neural networks, including CNNs, providing a solid theoretical basis for recent findings on fibration symmetries in geometric deep learning.