揭示神经网络推理的几何瓶颈，指出线性传播假设存在根本缺陷
Dynamics Reveals Structure: Challenging the Linear Propagation Assumption
- 用关系代数分析参数更新如何传递逻辑关系
- 发现组合操作导致特征映射坍塌，破坏推理一致性
- 为知识编辑失败和多跳推理难题提供结构层面解释
神经网络通过一阶参数更新进行适应，但这些更新是否保持逻辑连贯仍不明确。本文研究线性传播假设（LPA）的几何边界，即局部更新能否一致传播至逻辑推论。采用关系代数，考察三种核心关系操作：否定反转真值、对换交换参数顺序、复合串联关系。对否定和对换，证明方向无关的一阶传播需张量分解分离实体对上下文与关系内容；但对复合，发现根本性障碍：复合退化为合取，且在线性特征上定义良好的合取必为双线性，而双线性与否定不兼容，强制特征映射坍塌。结果表明，知识编辑失败、反转诅咒及多跳推理困境可能源于LPA固有的结构性限制。
原文摘要 · Abstract (English)
Neural networks adapt through first-order parameter updates, yet it remains unclear whether such updates preserve logical coherence. We investigate the geometric limits of the Linear Propagation Assumption (LPA), the premise that local updates coherently propagate to logical consequences. To formalize this, we adopt relation algebra and study three core operations on relations: negation flips truth values, converse swaps argument order, and composition chains relations. For negation and converse, we prove that guaranteeing direction-agnostic first-order propagation necessitates a tensor factorization separating entity-pair context from relation content. However, for composition, we identify a fundamental obstruction. We show that composition reduces to conjunction, and prove that any conjunction well-defined on linear features must be bilinear. Since bilinearity is incompatible with negation, this forces the feature map to collapse. These results suggest that failures in knowledge editing, the reversal curse, and multi-hop reasoning may stem from common structural limitations inherent to the LPA.
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