[2604.07401] Geometric Entropy and Retrieval Phase Transitions in Continuous Thermal Dense Associative Memory

Condensed Matter > Disordered Systems and Neural Networks arXiv:2604.07401 (cond-mat) [Submitted on 8 Apr 2026 (v1), last revised 6 May 2026 (this version, v2)] Title:Geometric Entropy and Retrieval Phase Transitions in Continuous Thermal Dense Associative Memory Authors:Tatiana Petrova, Evgeny Polyachenko, Radu State View a PDF of the paper titled Geometric Entropy and Retrieval Phase Transitions in Continuous Thermal Dense Associative Memory, by Tatiana Petrova and 2 other authors View PDF HTML (experimental) Abstract: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 $M = e^{\alpha N}$, 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 $\alpha = 0.5$ 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 $\alpha > 0$. For LSR, the finite support introduces a threshold $\alpha_{\text{th}}$ below which no spurious patterns contribute to the noise flo...