发表机构
Higher Institute of Informatics and Mathematics of Monastir, University of Monastir（蒙斯塔尔高等信息与数学学院，蒙斯塔尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究将深度学习中反向传播的数学结构用F-伴随框架重新表述为线性系统，证明全局反向算子幂零，揭示其与块回代的等价性，通过数值示例展示在不同架构中的表现，并推导残差网络和迁移学习机制，提升反向传播理论。
AI 中文摘要
反向传播是深度学习的计算引擎，但其数学结构通常被视为计算图的过程性遍历。我们提出了F-伴随框架的全局算子理论，将L层前馈网络的逐层反向递归重新表述为单个线性系统\((I - \cB)\Xs = \bG\)，其中\(\bG\)是源向量。我们证明全局反向算子\(\cB\)是严格块上三角且幂零指数至多为L。这种幂零性保证了诺伊曼级数解至多L项后精确终止，揭示经典反向传播在数学上等同于上双对角系统的块回代。我们形式化了F-对称性，通过数值示例展示了该算子视角如何揭示严格前馈网络的单路径坍缩及其在残差架构中的失效。最后，利用此组合结构严格推导了残差网络和迁移学习的机制。该框架将反向传播从算法配方提升为全局幂零算子公式。
英文摘要
Backpropagation is the computational engine of deep learning, yet its mathematical structure is typically treated as a procedural traversal of computational graphs. We present a global operator theory of the \emph{F-adjoint} framework, which reformulates the layerwise backward recursion of an $L$-depth feedforward network into a single linear system $(I-\cB)\Xs=\bG$, where $\bG$ is a source vector. We prove that the global backward operator $\cB$ is strictly block upper-triangular and nilpotent of index at most $L$. This nilpotency guarantees the exact termination of the Neumann series solution after at most $L$ terms, revealing classical backpropagation to be mathematically equivalent to block back-substitution on an upper bidiagonal system. We formalise \emph{F-symmetry} -- the condition in which the backward pass perfectly mirrors the forward pass -- identifying orthogonal weight matrices as canonical examples. Through worked numerical examples, we demonstrate how this operator perspective exposes the single-path collapse of strictly feedforward networks and its breakdown in residual architectures. Finally, we leverage this compositional structure to rigorously derive the mechanics of residual networks (gradient highways) and transfer learning (gradient truncation). This framework elevates backpropagation from an algorithmic recipe to a global nilpotent-operator formulation.
Comments23 pages, 3 figures
