Directional Non-Commutative Monoidal Structures for Compositional Embeddings in Machine Learning

TL;DR

This paper introduces a novel algebraic framework for multi-dimensional compositional embeddings using directional non-commutative monoidal operators, enabling structured, consistent, and flexible representations in machine learning.

Contribution

It presents the first unified multi-dimensional algebraic structure that generalizes classical models like SSMs and transformers, with formal properties and potential applications.

Findings

Defines axis-specific composition operators with associative properties

Ensures global consistency through commuting operators and interchange law

Provides a theoretical foundation for structured multi-dimensional embeddings

Abstract

We introduce a new algebraic structure for multi-dimensional compositional embeddings, built on directional non-commutative monoidal operators. The core contribution of this work is this novel framework, which exhibits appealing theoretical properties (associativity along each dimension and an interchange law ensuring global consistency) while remaining compatible with modern machine learning architectures. Our construction defines a distinct composition operator circ_i for each axis i, ensuring associative combination along each axis without imposing global commutativity. Importantly, all axis-specific operators commute with one another, enforcing a global interchange law that enables consistent crossaxis compositions. This is, to our knowledge, the first approach that provides a common foundation that generalizes classical sequence-modeling paradigms (e.g., structured state-space models (SSMs) and transformer self-attention) to a unified multi-dimensional framework. For example, specific one-dimensional instances of our framework can recover the familiar affine transformation algebra, vanilla self-attention, and the SSM-style recurrence. The higher-dimensional generalizations naturally support recursive, structure-aware operations in embedding spaces. We outline several potential applications unlocked by this structure-including structured positional encodings in Transformers, directional image embeddings, and symbolic modeling of sequences or grids-indicating that it could inform future deep learning model designs. We formally establish the algebraic properties of our framework and discuss efficient implementations. Finally, as our focus is theoretical, we include no experiments here and defer empirical validation to future work, which we plan to undertake.