Graded Neural Networks

This paper presents a novel framework for graded neural networks (GNNs) built over graded vector spaces _ⁿ, extending classical neural architectures by incorporating algebraic grading. Leveraging a coordinate-wise grading structure with scalar action λ = (λqᵢ xᵢ), defined by a tuple = (q₀, …, qₙ₋₁), we introduce graded neurons, layers, activation functions, and loss functions that adapt to feature significance. Theoretical properties of graded spaces are established, followed by a comprehensive GNN design, addressing computational challenges like numerical stability and gradient scaling. Potential applications span machine learning and photonic systems, exemplified by high-speed laser-based implementations. This work offers a foundational step toward graded computation, unifying mathematical rigor with practical potential, with avenues for future empirical and hardware exploration.