用测地线流匹配修复高维符号表示的几何误差
Geodesic Flow Matching for Denoising High-Dimensional Structured Representations
- 在环面流形上使用测地线流匹配，确保去噪过程不破坏符号结构
- 在神经导航系统中减少72%追踪误差，提升40%神经效率
- 适合需要精确符号推理的神经符号系统研究者
向量符号代数（VSAs）通过将符号信息编码为高维分布式表示，实现鲁棒的神经符号推理。针对连续域，空间语义指针（SSPs）将变量映射到连续环面流形上。然而，传统流匹配方法假设欧氏平坦几何，无法满足SSP有效状态的几何约束。我们发现，欧氏线性插值会“穿过”流形内部，破坏解码所需的相位和幅值结构。为此，提出测地线流匹配，利用黎曼传输动力学严格限制去噪流在SSP环面流形上。在脉冲神经SLAM系统中验证，该方法使路径积分对漂移更稳定，追踪误差降低72%，神经效率提升40%。代码已开源。
原文摘要 · Abstract (English)
Vector Symbolic Algebras (VSAs) enable robust neurosymbolic reasoning by encoding symbolic information into high-dimensional distributed representations. For continuous domains, Spatial Semantic Pointers (SSPs) extend this framework by mapping variables onto continuous toroidal manifolds. However, standard approaches like Flow Matching assume a flat Euclidean geometry, which fails to account for the geometric constraints imposed on valid SSP states. We demonstrate that this assumption fails for SSPs: Euclidean linear interpolants ``cut through" the manifold's interior, destroying the phase and magnitude structure required for accurate decoding. To resolve this, we employ Geodesic Flow Matching, adapting Riemannian transport dynamics to strictly restrict the denoising flow to the SSP toroidal manifold. We validate this approach in a Spiking Neural SLAM system, showing that manifold-aware cleanup stabilizes path integration against drift. The method achieves a 72\% reduction in tracking error and enables a 40\% increase in neural efficiency compared to competitive baselines. Code is available at https://github.com/kremHabashy/CleanupSSP .
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