统一傅里叶、哈达玛等变换的高维代数框架
Directional Non-Commutative Monoidal Structures with Interchange Law via Commutative Generators
- 用可交换生成元递归构建非交换多维复合结构
- 将DFT、Walsh、Hadamard等变换纳入统一框架
- 支持针对特定数据学习定制化变换
我们提出一种新型代数框架，通过定义满足非交换性与全局交换律的轴向复合算子，将一维单幺半群系统推广至高维空间。该结构从向量-矩阵对的基态递归定义，建模多维方向性复合，同时利用可交换线性算子保持结构一致性。该框架统一了信号处理与数据分析中的多种经典线性变换。数据索引被嵌入复合结构并可分解为更简成分。我们证明离散傅里叶变换（DFT）、沃尔什变换和哈达玛变换均为该代数结构的特例。通过合理选择向量与矩阵对，可系统推导这些变换。该框架不仅整合经典变换，还支持针对特定数据模态与任务设计可学习变换。
原文摘要 · Abstract (English)
We introduce a novel framework consisting of a class of algebraic structures that generalize one-dimensional monoidal systems into higher dimensions by defining per-axis composition operators subject to non-commutativity and a global interchange law. These structures, defined recursively from a base case of vector-matrix pairs, model directional composition in multiple dimensions while preserving structural coherence through commutative linear operators. We show that the framework that unifies several well-known linear transforms in signal processing and data analysis. In this framework, data indices are embedded into a composite structure that decomposes into simpler components. We show that classic transforms such as the Discrete Fourier Transform (DFT), the Walsh transform, and the Hadamard transform are special cases of our algebraic structure. The framework provides a systematic way to derive these transforms by appropriately choosing vector and matrix pairs. By subsuming classical transforms within a common structure, the framework also enables the development of learnable transformations tailored to specific data modalities and tasks.
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