Graded Neural Networks

• URL: http://arxiv.org/abs/2502.17751v2
• Date: Sat, 05 Jul 2025 22:36:13 GMT
• Title: Graded Neural Networks
• Authors: Tony Shaska,
• Abstract summary: This paper presents a novel framework for graded neural networks (GNNs) built over graded vector spaces.<n>Potential applications span machine learning and photonic systems, exemplified by high-speed laser-based implementations.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\lambda \star \x = (\lambda^{q_i} x_i)$, 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.

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