Weaves, Wires, and Morphisms: Formalizing and Implementing the Algebra of Deep Learning
This work addresses the problem of ad-hoc notation and poor handling of nonlinear broadcasting in deep learning for researchers and practitioners, laying a foundational framework for systematic model design and analysis, though it is incremental in formalizing existing concepts.
The paper tackles the lack of a formal mathematical framework for describing deep learning model architectures by introducing a categorical framework that formalizes broadcasting through axis-stride and array-broadcasted categories, enabling precise expression and manipulation of architectures in a compositional manner. It provides implementations in Python and TypeScript to demonstrate universal applicability, including features like algebraic construction, graph conversion, PyTorch compilation, and diagram rendering.
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.