arXiv：cs.LG· Hyunsang Hwang, Suhyun Bae, Donghun Lee·· 7 小时前AI 评分36
Prime Fourier Embeddings：面向模运算的原理化基
Prime Fourier Embeddings: A Principled Basis for Modular Arithmetic
AI 导读
研究者提出 Prime Fourier Embeddings（PFE），将整数编码为源自 Q 调和分析的素数索引 (cos, sin) 对，使模运算只需选择相关的素数通道。
正文
Abstract:Numbers have algebraic structure that standard neural embeddings often fail to expose. We introduce Prime Fourier Embeddings (PFE), which encode integers as prime-indexed (cos, sin) pairs derived from the harmonic analysis of Q, providing a pre-structured representation in which modular arithmetic reduces to selecting the relevant prime channel rather than discovering algebraic structure from scratch. We prove that any linear map equivariant with respect to the product group action on PFE must be block-diagonal with one independent block per prime -- a consequence of Schur's lemma applied to the resulting character decomposition. For square-free composite moduli, the Chinese Remainder Theorem predicts which prime channels are task-relevant. Both predictions are confirmed empirically: ablation studies show specialization ratios exceeding 500x between task-relevant and task-irrelevant channels, with perfect in-distribution test accuracy across all square-free composite moduli tested.
| Subjects: | Machine Learning (cs.LG); Artificial Intelligence (cs.AI) |
| Cite as: | arXiv:2606.23044 [cs.LG] |
| (or arXiv:2606.23044v3 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2606.23044 arXiv-issued DOI via DataCite |
Submission history
From: Hyunsang Hwang [view email]
[v1]
Mon, 22 Jun 2026 08:50:41 UTC (1,927 KB)
[v2]
Tue, 14 Jul 2026 05:47:21 UTC (1,764 KB)
[v3]
Tue, 6 Oct 2026 07:33:00 UTC (1,726 KB)
来源：arXiv：cs.LG · arxiv.org