从对称性到不变性：学习有限域中的伽罗瓦等价表示
From Symmetry to Invariance: Learning Galois Equivalent Representations in Finite Fields
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中文总结 AI 辅助
该研究针对神经网络在有限域运算等价表示间的迁移问题，提出训练模型预测伽罗瓦作用以构造轨道规范代表的方法，实现了乘法运算向未见过的基表示的迁移。
中文摘要 AI 辅助
神经网络可从有限样本中学习代数运算，但该能力能否在同一运算的数学等价表示间迁移仍不明确。我们通过有限域中基变换下的乘法研究该问题：伽罗瓦作用将基表示组织为轨道，同一轨道内的基会诱导相同的坐标乘法映射，该结构可使我们将乘法学习与该乘法向训练未使用的基表示的迁移分离开。我们考察了提供或恢复相关轨道结构的多种方式，包括不变标签、基矩阵、轨道识别及代数分解。我们的主要方法是训练模型以预测基表示间的伽罗瓦作用，随后利用所学变换的重复应用为每个轨道构造一个规范代表，该代表通过精确的规范匹配支持对保留基的乘法运算，这为将所学代数对称性转换为可用于迁移的不变表示提供了具体机制。
英文摘要
Neural networks can learn algebraic operations from finite examples, but it remains unclear whether this ability transfers across mathematically equivalent representations of the same operation. We study this question through multiplication in finite fields under changes of basis. The Galois action organizes basis representations into orbits, and bases in the same orbit induce the same coordinate multiplication map. This structure allows us to separate learning multiplication from transferring it to basis representations that are not used for training. We examine several ways of providing or recovering the relevant orbit structure, including invariant labels, basis matrices, orbit recognition, and algebraic decomposition. Our main approach trains a model to predict the Galois action between basis representations. Repeated applications of the learned transformation are then used to construct a canonical representative for each orbit, which supports multiplication on held-out bases through exact canonical matching. This provides a concrete mechanism for converting a learned algebraic symmetry into an invariant representation that can be used for transfer.
发表机构
- Towson University（陶森大学）
机构由 AI 辅助整理，请以论文原文为准。
