The algebra and the geometry aspect of Deep learning

TL;DR

This paper explores the geometric and algebraic foundations of deep learning, emphasizing the role of differential calculus, manifold structures, and homotopy theory in understanding neural network behavior and interpretability.

Contribution

It introduces a coordinate-free formulation of backpropagation, reinterprets classification through geometric tools, and connects neural networks to algebraic topology concepts.

Findings

Backpropagation derived from scalar product on matrix spaces.

Manifold and algebraic topology tools inform convergence and interpretability.

Neural networks modeled via directed graphs and homotopy theory.

Abstract

This paper investigates the foundations of deep learning through insight of geometry, algebra and differential calculus. At is core, artificial intelligence relies on assumption that data and its intrinsic structure can be embedded into vector spaces allowing for analysis through geometric and algebraic methods. We thrace the development of neural networks from the perceptron to the transformer architecture, emphasizing on the underlying geometric structures and differential processes that govern their behavior. Our original approach highlights how the canonical scalar product on matrix spaces naturally leads to backpropagation equations yielding to a coordinate free formulation. We explore how classification problems can reinterpreted using tools from differential and algebraic geometry suggesting that manifold structure, degree of variety, homology may inform both convergence and interpretability of learning algorithms We further examine how neural networks can be interpreted via their associated directed graph, drawing connection to a Quillen model defined in [1] and [13] to describe memory as an homotopy theoretic property of the associated network.