Sinkhorn Based Associative Memory Retrieval Using Spherical Hellinger Kantorovich Dynamics

Dense associative memory underlies modern attention mechanisms; however, Modern Hopfield Networks (MHNs) with linear kernels yield overlapping attractor basins and spurious states under high memory load, limiting their theoretically exponential storage capacity in practice. We construct a class of nonlinear phase kernels that modify the energy landscape and retrieval dynamics, and prove that these kernels preserve exponential storage capacity and local fixed-point convergence under given conditions on kernel smoothness, memory separation, and memory distributions. The framework is instantiated by the Sigmoidal Phase Hopfield Network (S-Hop), which guarantees monotone energy descent and mitigates practical capacity loss. S-Hop has $8.3\times$ and $4\times$ gains in one-step noisy retrieval load over Uniform Hopfield Networks (U-Hop) on CIFAR-10 and TinyImageNet respectively, and reaches a $50\times$ gain in first-failure load for ImageNet. The results demonstrate that phase kernels provide a principled route to robust dense associative memory.

Geometry does more than constrain an associative memory: curvature determines what it remembers and which states it creates. We develop intrinsic dense associative memories on Riemannian manifolds by casting memory as Epanechnikov kernel-density mode seeking. We compare geodesic and volume-corrected energies and show that curvature separates their behavior. We prove that geodesic memory always retains an isolated pattern, while corrected memory obeys a sharp Ricci-curvature threshold: positive curvature can erase memories in high dimensions, while negative curvature reinforces them. We derive geodesic capacity scalings of $q_β^{-1/2}$ for retaining every pattern and $q_β^{-1}$ for a typical one, where $q_β$ is the pairwise kernel-overlap probability. We show how overlap creates novel memories: designed $N$-pattern configurations realize all $2^N-1$ subset modes, but random data at the storage threshold yield only a Poisson number. We establish exact one-step recall using Riemannian mean shift. In simulations, we recover the predicted curvature transition and every designed mode. On WordNet's full noun hierarchy, we demonstrate that volume correction improves low-capacity retrieval. Together, our work shows that curvature is a design variable for associative memory, not merely a property of the data.

We study the thermodynamic memory capacity of modern Hopfield networks (Dense Associative Memory models) with continuous states under geometric constraints, extending classical analyses of pairwise associative memory. We derive thermodynamic phase boundaries for Dense Associative Memory networks with exponential capacity $M = e^{αN}$, comparing Gaussian (LSE) and Epanechnikov (LSR) kernels. For continuous neurons on an $N$-sphere, the geometric entropy depends solely on the spherical geometry, not the kernel. In the sharp-kernel regime, the maximum theoretical capacity $α= 0.5$ is achieved at zero temperature; below this threshold, a critical line separates retrieval from non-retrieval. The two kernels differ qualitatively in their phase boundary structure: for LSE, a critical line exists at all loads $α> 0$. For LSR, the finite support introduces a threshold $α_{\text{th}}$ below which no spurious patterns contribute to the noise floor, and no critical line exists -- retrieval is perfect at any temperature. These results advance the theory of high-capacity associative memory and clarify fundamental limits of retrieval robustness in modern attention-like memory architectures.