Directional Non-Commutative Monoidal Structures with Interchange Law via Commutative Generators

TL;DR

This paper introduces a new algebraic framework that generalizes classical linear transforms like DFT, Walsh, and Hadamard into a unified, higher-dimensional structure with directional composition and interchange laws, enabling tailored data transformations.

Contribution

The paper presents a novel algebraic framework that unifies classical transforms and allows for the development of learnable, task-specific transformations in data analysis.

Findings

Classical transforms are special cases of the proposed framework.

The framework models directional, multi-dimensional composition with interchange laws.

It enables systematic derivation of transforms and customization for specific data modalities.

Abstract

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.