# Geodesic Flow Matching for Denoising High-Dimensional Structured Representations

arXiv: https://arxiv.org/abs/2606.00248 · HTML: https://mlsanity.com/paper/2606.00248/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning.
- Builds on 10 of its 21 references that are in the map.
- First posted on arXiv 29 May 2026.

## Placed in

- LLM reasoning / [Spatial and embodied reasoning](https://mlsanity.com/lr/spatial-and-embodied-reasoning/)
- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 10 of 10)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [Hyperspherical Prototype Networks](https://mlsanity.com/paper/1901.10514/) (2019)
- [A comparison of Vector Symbolic Architectures](https://mlsanity.com/paper/2001.11797/) (2020)
- [Riemannian Flow Matching Policy for Robot Motion Learning](https://mlsanity.com/paper/2403.10672/) (2024)
- [Memorization to Generalization: Emergence of Diffusion Models from Associative Memory](https://mlsanity.com/paper/2505.21777/) (2025)
- [Memory in Plain Sight: Surveying the Uncanny Resemblances of Associative Memories and Diffusion Models](https://mlsanity.com/paper/2309.16750/) (2023)
- [Binding in hippocampal-entorhinal circuits enables compositionality in cognitive maps](https://mlsanity.com/paper/2406.18808/) (2024)

## Built on by (most cited first, 1 of 1)

- [Manifold-Stable Flow Matching](https://mlsanity.com/paper/2609.35454/) (2026)

## Abstract (arXiv)

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 .