Geodesic Flow Matching for Denoising High-Dimensional Structured Representations

Vector Symbolic Algebras (VSAs) enable robust neurosymbolic reasoning by encoding information into high-dimensional distributed representations. For continuous domains, Spatial Semantic Pointers (SSPs) extend this framework by mapping variables onto precise toroidal manifolds. While generative models offer a promising avenue for cleaning up (denoising) these representations, standard approaches like Flow Matching assume a flat Euclidean geometry. 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 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 classical baselines.

AI systems that combine neural networks with symbolic reasoning encode information as points on the surface of a high-dimensional sphere. Over time, noise pushes these points off the surface, so a cleanup step is needed to restore them. Standard generative AI cleanup methods assume space is flat and move points along straight lines, but straight lines between two points on a sphere pass through the interior, destroying the encoded information in the process. We fix this by making the cleanup travel along the curved surface of the sphere instead. We tested this in a robotic navigation task where a simulated agent must track its location and build a map of its surroundings using brain-inspired spiking neural networks. Our method reduces navigation error by 72% and achieves the same accuracy with 40% fewer neurons. This matters because spiking neural networks are the basis for energy-efficient neuromorphic chips, and making them work reliably with less hardware is a key step toward deploying intelligent systems in resource-constrained environments like robots and edge devices.