Sinkhorn Based Associative Memory Retrieval Using Spherical Hellinger Kantorovich Dynamics
Abstract
We propose a dense associative memory for empirical measures (weighted point clouds). Stored patterns and queries are finitely supported probability measures, and retrieval is defined by minimizing a Hopfield-style log-sum-exp energy built from the debiased Sinkhorn divergence. We derive retrieval dynamics as a spherical Hellinger Kantorovich (SHK) gradient flow, which updates both support locations and weights. Discretizing the flow yields a deterministic algorithm that uses Sinkhorn potentials to compute barycentric transport steps and a multiplicative simplex reweighting. Under local separation and PL-type conditions we prove basin invariance, geometric convergence to a local minimizer, and a bound showing the minimizer remains close to the corresponding stored pattern. Under a random pattern model, we further show that these Sinkhorn basins are disjoint with high probability, implying exponential capacity in the ambient dimension. Experiments on synthetic Gaussian point-cloud memories demonstrate robust recovery from perturbed queries versus a Euclidean Hopfield-type baseline.
Cite
@article{arxiv.2603.20656,
title = {Sinkhorn Based Associative Memory Retrieval Using Spherical Hellinger Kantorovich Dynamics},
author = {Aratrika Mustafi and Soumya Mukherjee},
journal= {arXiv preprint arXiv:2603.20656},
year = {2026}
}