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.

We present a novel algorithm to estimate the barycenter of arbitrary probability distributions with respect to the Sinkhorn divergence. Based on a Frank-Wolfe optimization strategy, our approach proceeds by populating the support of the barycenter incrementally, without requiring any pre-allocation.

We thank the reviewers for their comments. We then address reviewer's comments individually (due to space limits please zoom in the tiny figures). For [18] we used Alg. 2 We thank the reviewer for the additional reference, which we will add to the paper. Gradient Descent) applied in parallel to multiple starting points. We thank R2 for the reference "Entropic regularization of continuous optimal transport problems".

Assume the conditions in Proposition 1 hold.

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Efficiently aggregating data from different sources is a challenging problem, particularly when samples from each source are distributed differently.

We present a novel algorithm to estimate the barycenter of arbitrary probability distributions with respect to the Sinkhorn divergence. Based on a Frank-Wolfe optimization strategy, our approach proceeds by populating the support of the barycenter incrementally, without requiring any pre-allocation.

We thank the reviewers for their comments. We then address reviewer's comments individually (due to space limits please zoom in the tiny figures). For [18] we used Alg. 2 We thank the reviewer for the additional reference, which we will add to the paper. Gradient Descent) applied in parallel to multiple starting points. We thank R2 for the reference "Entropic regularization of continuous optimal transport problems".