Sinkhorn Based Associative Memory Retrieval Using Spherical Hellinger Kantorovich Dynamics
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| Format: | Preprint |
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2026
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| _version_ | 1866908904924905472 |
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| author | Mustafi, Aratrika Mukherjee, Soumya |
| author_facet | Mustafi, Aratrika Mukherjee, Soumya |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_20656 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sinkhorn Based Associative Memory Retrieval Using Spherical Hellinger Kantorovich Dynamics Mustafi, Aratrika Mukherjee, Soumya Machine Learning Artificial Intelligence Optimization and Control Statistics Theory 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. |
| title | Sinkhorn Based Associative Memory Retrieval Using Spherical Hellinger Kantorovich Dynamics |
| topic | Machine Learning Artificial Intelligence Optimization and Control Statistics Theory |
| url | https://arxiv.org/abs/2603.20656 |