让神经网络学会在有限域中跨基变换迁移乘法运算
From Symmetry to Invariance: Learning Galois Equivalent Representations in Finite Fields
- 通过学习伽罗瓦作用，构建基变换下的不变表示
- 在未训练的基上实现精确乘法匹配，准确率接近100%
- 适合对代数结构与模型泛化感兴趣的研究者
神经网络可从有限样本中学习代数运算，但其能力能否在数学等价的不同基表示间迁移尚不明确。本文以有限域中的乘法为例，研究基变换下的迁移问题。伽罗瓦作用将基划分为轨道，同一轨道内的基诱导相同的坐标乘法映射。我们探索了多种获取或恢复轨道结构的方法，包括不变标签、基矩阵、轨道识别和代数分解。核心方法是训练模型预测基表示间的伽罗瓦作用。通过重复应用学习到的变换，构造每条轨道的规范代表元，从而在未见基上通过精确规范匹配实现乘法。该机制为将学习到的代数对称性转化为可迁移的不变表示提供了具体方案。
原文摘要 · Abstract (English)
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.
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