Geodesic Flow Matching for Denoising High-Dimensional Structured Representations

--> Computer Science > Artificial Intelligence arXiv:2606.00248 (cs) [Submitted on 29 May 2026] Title: Geodesic Flow Matching for Denoising High-Dimensional Structured Representations Authors: Karim Habashy , Chris Eliasmith View a PDF of the paper titled Geodesic Flow Matching for Denoising High-Dimensional Structured Representations, by Karim Habashy and Chris Eliasmith View PDF HTML (experimental) Abstract: 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 this https URL . Comments: ICML 2026 Main track Subjects: Artificial Intelligence (cs.AI) Cite as: arXiv:2606.00248 [cs.AI] (or arXiv:2606.00248v1 [cs.AI] for this version) https://doi.org/10.48550/arXiv.2606.00248 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Karim Habashy [ view email ] [v1] Fri, 29 May 2026 18:28:41 UTC (6,048 KB) Full-text links: Access Paper: View a PDF of the paper titled Geodesic Flow Matching for Denoising High-Dimensional Structured Representations, by Karim Habashy and Chris Eliasmith View PDF HTML (experimental) TeX Source view license Current browse context: cs.AI < prev | next > new | recent | 2026-06 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? )