Sinkhorn Based Associative Memory Retrieval Using Spherical Hellinger Kantorovich Dynamics

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.

Convex Smoothed Autoencoder-Optimal Transport model

Generative modelling is a key tool in unsupervised machine learning which has achieved stellar success in recent years. Despite this huge success, even the best generative models such as Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs) come with their own shortcomings, mode collapse and mode mixture being the two most prominent problems. In this paper we develop a new generative model capable of generating samples which resemble the observed data, and is free from mode collapse and mode mixture. Our model is inspired by the recently proposed Autoencoder-Optimal Transport (AE-OT) model and tries to improve on it by addressing the problems faced by the AE-OT model itself, specifically with respect to the sample generation algorithm. Theoretical results concerning the bound on the error in approximating the non-smooth Brenier potential by its smoothed estimate, and approximating the discontinuous optimal transport map by a smoothed optimal transport map estimate have also been established in this paper.