The algebra and the geometry aspect of Deep learning

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.