{"ID":2888654,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.22396","arxiv_id":"2507.22396","title":"CLuP practically achieves $\\sim 1.77$ positive and $\\sim 0.33$ negative Hopfield model ground state free energy","abstract":"We study algorithmic aspects of finding $n$-dimensional \\emph{positive} and \\emph{negative} Hopfield ($\\pm$Hop) model ground state free energies. This corresponds to classical maximization of random positive/negative semi-definite quadratic forms over binary $\\left \\{\\pm \\frac{1}{\\sqrt{n}} \\right \\}^n$ vectors. The key algorithmic question is whether these problems can be computationally efficiently approximated within a factor $\\approx 1$. Following the introduction and success of \\emph{Controlled Loosening-up} (CLuP-SK) algorithms in finding near ground state energies of closely related Sherrington-Kirkpatrick (SK) models [82], we here propose a CLuP$\\pm$Hop counterparts for $\\pm$Hop models. Fully lifted random duality theory (fl RDT) [78] is utilized to characterize CLuP$\\pm$Hop \\emph{typical} dynamics. An excellent agreement between practical performance and theoretical predictions is observed. In particular, for $n$ as small as few thousands CLuP$\\pm$Hop achieve $\\sim 1.77$ and $\\sim 0.33$ as the ground state free energies of the positive and negative Hopfield models. At the same time we obtain on the 6th level of lifting (6-spl RDT) corresponding theoretical thermodynamic ($n\\rightarrow\\infty$) limits $\\approx 1.7784$ and $\\approx 0.3281$. This positions determining Hopfield models near ground state energies as \\emph{typically} easy problems. Moreover, the very same 6th lifting level evaluations allow to uncover a fundamental intrinsic difference between two models: $+$Hop's near optimal configurations are \\emph{typically close} to each other whereas the $-$Hop's are \\emph{typically far away}.","short_abstract":"We study algorithmic aspects of finding $n$-dimensional \\emph{positive} and \\emph{negative} Hopfield ($\\pm$Hop) model ground state free energies. This corresponds to classical maximization of random positive/negative semi-definite quadratic forms over binary $\\left \\{\\pm \\frac{1}{\\sqrt{n}} \\right \\}^n$ vectors. The key...","url_abs":"https://arxiv.org/abs/2507.22396","url_pdf":"https://arxiv.org/pdf/2507.22396v1","authors":"[\"Mihailo Stojnic\"]","published":"2025-07-30T05:30:46Z","proceeding":"cond-mat.dis-nn","tasks":"[\"cond-mat.dis-nn\",\"cs.IT\",\"math.OC\",\"stat.ML\"]","methods":"[]","has_code":false}