用代数分级结构改进神经网络，让不同特征重要性可调。
Graded Neural Networks
- 基于标量作用的分级向量空间设计新型神经元与层
- 实现梯度稳定与计算可扩展性，支持高速光子系统部署
- 适合对数学严谨性与硬件效率有要求的研究者
本文提出一种在分级向量空间 $\V_\w^n$ 上构建的分级神经网络（GNN）新框架，通过坐标级的标量作用 $λ\star \x = (λ^{q_i} x_i)$ 及参数 $\w = (q_0, \ldots, q_{n-1})$ 引入分级机制，使神经元、层、激活函数和损失函数能响应特征重要性。理论分析建立了分级空间性质，并完成了完整的 GNN 设计，解决数值稳定性与梯度缩放等计算挑战。应用涵盖机器学习与光子系统，例如高速激光实现。该工作为分级计算奠定基础，融合数学严谨性与实际潜力，未来可开展实证与硬件探索。
原文摘要 · Abstract (English)
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.
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