Sinkhorn based Associative Memory retrieval using Spherical Hellinger Kantorovich dynamics

Aratrika Mustafi, Soumya Mukherjee

Keywords: Associative Memory retrieval, spherical Hellinger Kantorovich, log-sum-exp, Sinkhorn divergence

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 baseline.

Submission Number: 54