Representation Redundancy and Structural Complexity in Finite-Field Inversion
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Abstract (arXiv)
The representation chosen for a mathematical operation can affect both its algebraic form and its empirical learning difficulty. We study this phenomenon for inversion over (mathbb F2n), with field elements expressed in varying ordered (mathbb F2)-bases. We prove that two ordered bases induce the same coordinate inversion map if and only if they belong to the same Galois orbit. Since every orbit has size (n), the correspondence between ordered bases and distinct inversion maps is exactly (n)-to-one. We then analyze three Boolean formulations of inversion. The reference formulation has algebraic degree (n-1) and joint ANF leap (1), the mixed representation formulation has degree (2(n-1)) and joint ANF leap (2), and the complete raw formulation has degree at most (3(n-1)) and joint ANF leap at least (n). Exhaustive computations agree with the theoretical results and bounds in the cases considered. Controlled experiments with multilayer perceptrons show the same ordering in learning difficulty, while Galois orbit redundancy provides only a limited generalization benefit under the tested conditions. These results show that exact redundancy among representations can coexist with changes in Boolean structure and learning behavior when the representation is exposed as part of the input.
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References · 12
- 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
- 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,167
- 2016Group Equivariant Convolutional Networkscited by 1,165
- 2023SGD learning on neural networks: leap complexity and saddle-to-saddle dynamicscited by 143
- 2022The merged-staircase property: a necessary and nearly sufficient condition for SGD learning of sparse functions on two-layer neural networkscited by 142
- 2021Provably Strict Generalisation Benefit for Equivariant Modelscited by 91
- 2016Generalization Error of Invariant Classifierscited by 42
- 2020Theoretical Aspects of Group Equivariant Neural Networkscited by 25
- 2021On the Sample Complexity of Learning under Invariance and Geometric Stabilitycited by 13
- 2024Learning to Understand: Identifying Interactions via the M"obius Transformcited by 13
- 2024Role of Locality and Weight Sharing in Image-Based Tasks: A Sample Complexity Separation between CNNs, LCNs, and FCNscited by 5
- 2026From Symmetry to Invariance: Learning Galois Equivalent Representations in Finite Fieldscited by 1
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