Title: Geodesic Flow Matching for Denoising High-Dimensional Structured Representations

Abstract

Vector Symbolic Algebras (VSAs) facilitate robust neurosymbolic reasoning by embedding symbolic data into high-dimensional distributed representations. Within continuous domains, Spatial Semantic Pointers (SSPs) expand this paradigm by projecting variables onto continuous toroidal manifolds. Conventional methods, such as standard Flow Matching, operate under the assumption of flat Euclidean geometry, thereby neglecting the geometric constraints inherent to valid SSP states. We illustrate that this Euclidean assumption is inadequate for SSPs; specifically, linear interpolants in Euclidean space traverse the interior of the manifold, thereby disrupting the phase and magnitude structures essential for precise decoding. To address this limitation, we implement Geodesic Flow Matching, which adapts Riemannian transport dynamics to confine the denoising process strictly within the SSP toroidal manifold. We evaluated this technique within a Spiking Neural SLAM framework, demonstrating that manifold-aware denoising stabilizes path integration by mitigating drift. Our results indicate a 72% decrease in tracking error and a 40% improvement in neural efficiency relative to competitive baseline methods. The source code is accessible at https://github.com/kremHabashy/CleanupSSP .

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC