This is a title slide featuring the Chinese title "多项式展开对网络计算结构的影响." The English subtitle translates it as "Impact of Polynomial Expansion on Network Computational Structure."

多项式展开对网络计算结构的影响

Impact of Polynomial Expansion on Network Computational Structure

This introduction slide highlights the expressive limitations of traditional activation functions in breadth modeling. It proposes polynomial expansions as an alternative and aims to explore their theoretical support for breadth structures.

引言

• 问题背景：传统激活函数在广度建模中的表达障碍
• 解决思路：引入“多项式展开”作为替代表达形式
• 目标：探讨其对广度结构的理论支撑能力

Polynomials, formed by finite power functions and closed under addition and multiplication, naturally fit combinatorial expansion structures. This aligns with breadth networks through feature combinations and linear weighting, providing unified, composable, parallelizable, and hardware-friendly advantages.

代数封闭性——天然适配组合展开结构

• 多项式由有限幂函数构成，加法乘法封闭
• 适配广度网络：特征组合 + 线性加权
• 优势：统一、可组合、可并行、硬件友好

Source: 多项式展开对网络计算结构的影响

Polynomials offer smooth derivatives and stable gradients superior to ReLU and Sigmoid, while suppressing high-order oscillations like the Runge phenomenon. Chebyshev/Legendre basis functions enhance fitting ability, with regularization terms improving generalization and stability.

数值稳定性——光滑性与拟合能力

• 多项式导数光滑、梯度稳定（优于ReLU与Sigmoid）
• 抑制高阶振荡（Runge现象）
• Chebyshev/Legendre基函数提升拟合能力
• 正则项增强泛化与稳定性

This slide on "Expressible Parsability—Structural Reconstructability and Differentiability" emphasizes clear analytical expressions that are everywhere differentiable. It supports gradient propagation and backpropagation, enables DNN-to-BNN restructuring and flattening, and boosts feature crossing with model interpretability.

表达可解析——结构可重构与可导性

• 明确的解析表达式与处处可导性
• 支持梯度传播与误差回传
• DNN 到 BNN 结构重构与压平
• 增强特征交叉与模型可解释性

Source: 多项式具备明确的解析表达式与处处可导性。支持梯度传播与误差回传。实现从 DNN 到 BNN 的结构重构与压平，增强特征交叉与可解释性。

This slide highlights enhanced interpretability through clear polynomial expressions decomposable into input combination functions. It supports path visualization, symbolic reasoning, structural pruning, and improves model transparency and trustworthiness.

可解释性增强——符号表达与路径追踪

• 多项式表达结构清晰，可分解为输入组合函数
• 支持路径可视化、符号推理与结构裁剪
• 提升模型透明性与可信性

Polynomial expressions overcome the structural limitations of traditional activations. They provide structural expansion and parsing support for the D2B framework, serving as a key step in transforming depth to breadth.

结论

多项式表达克服传统激活的结构限制。 为 D2B 框架提供结构展开与解析支持。 是实现深度向广度转化的关键步骤。

Source: 多项式展开对网络计算结构的影响