Weaves, Wires, and Morphisms: Formalizing and Implementing the Algebra of Deep Learning

TL;DR

This paper introduces a formal categorical framework for deep learning architectures, enabling precise, compositional mathematical descriptions and implementations across multiple programming languages.

Contribution

It develops a novel categorical approach with axis-stride and array-broadcasted categories, translating mathematical models into diagrams and code for systematic deep learning design.

Findings

Framework formalizes broadcasting and model composition.

Provides Python and TypeScript implementations.

Enables algebraic construction, graph conversion, and diagram rendering.

Abstract

Despite deep learning models running well-defined mathematical functions, we lack a formal mathematical framework for describing model architectures. Ad-hoc notation, diagrams, and pseudocode poorly handle nonlinear broadcasting and the relationship between individual components and composed models. This paper introduces a categorical framework for deep learning models that formalizes broadcasting through the novel axis-stride and array-broadcasted categories. This allows the mathematical function underlying architectures to be precisely expressed and manipulated in a compositional manner. These mathematical definitions are translated into human manageable diagrams and machine manageable data structures. We provide a mirrored implementation in Python (pyncd) and TypeScript (tsncd) to show the universal aspect of our framework, along with features including algebraic construction, graph conversion, PyTorch compilation and diagram rendering. This lays the foundation for a systematic, formal approach to deep learning model design and analysis.