Graded Neural Networks

This paper presents a novel framework for graded neural networks (GNNs) built over graded vector spaces ${\text{V}}_{\text{w}}^n$, extending classical neural architectures by incorporating algebraic grading. Leveraging a coordinate-wise grading structure with scalar action $\lambda \star \text{x}=({\lambda }^{q_i}{x}_i)$, defined by a tuple $\text{w}=(q_0,\dots ,q_{n-1})$, 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.