Graded Neural Networks
This work provides a foundational step toward graded computation, potentially impacting machine learning and photonic systems, but it is incremental as it builds on existing neural architectures with new algebraic structures.
The paper tackles the problem of extending classical neural networks by incorporating algebraic grading into a novel framework called graded neural networks (GNNs), establishing theoretical properties and addressing computational challenges like numerical stability and gradient scaling.
This paper presents a novel framework for graded neural networks (GNNs) built over graded vector spaces $\V_\w^n$, extending classical neural architectures by incorporating algebraic grading. Leveraging a coordinate-wise grading structure with scalar action $λ\star \x = (λ^{q_i} x_i)$, defined by a tuple $\w = (q_0, \ldots, q_{n-1})$, 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.