Associative Memory for Hyperdimensional Spatial Representations with Geodesic Flow Matching

Abstract

The ability to recover clean patterns from noisy or partial inputs, known as Associative memory, is a cornerstone of robust computation in both biological and artificial intelligence systems. While well-established for discrete data, implementing this capability for continuous representations remains a challenge. Existing methods typically rely on discretizing the continuous domain into capacity-limited prototypes or performing explicit decoding to the spatial domain, which is often computationally expensive and biologically implausible.

This thesis addresses this gap by reformulating cleanup as a generative transport problem entirely within the embedding space. Geodesic Flow Matching is introduced as a model that learns a continuous time-dependent velocity field to transport corrupted representations back to the valid data manifold. Standard Euclidean Flow matching is shown to be insufficient for high-dimensional normalized representations, as linear interpolants "cut through" the interior of the hypersphere, destroying the vector magnitude and phase relationships required for accurate decoding. By constraining transport dynamics to the intrinsic Riemannian geometry of the hypersphere, the proposed model preserves this structure even under severe corruption.

The framework is validated using Spatial Semantic Pointers (SSPs), a biologically plausible encoding for continuous space. Benchmarks indicate that Geodesic Flow consistently outperforms Euclidean variants and classical baselines, particularly in high-noise regimes. The utility of the approach is further demonstrated through integration into a Spiking Neural SLAM system. As an online stabilizer for the path integrator, the Geodesic model prevents catastrophic drift, reducing path error by up to 72%. Furthermore, it significantly improves resource efficiency by 40%, allowing a neural population of 1,500 neurons to match the tracking accuracy of a baseline system using 2,500 neurons.