标题： CLuP 实际上实现了$\sim 1.77$个正和$\sim 0.33$个负的 Hopfield 模型基态自由能 显示英文标题

标题： CLuP practically achieves $\sim 1.77$ positive and $\sim 0.33$ negative Hopfield model ground state free energy

摘要： 我们研究寻找$n$维\emph{正的}和\emph{负的}Hopfield ($\pm$Hop) 模型基态自由能的算法方面。 这对应于在二进制$\left \{\pm \frac{1}{\sqrt{n}} \right \}^n$向量上对随机正/负半定二次型的经典最大化。 关键的算法问题是这些问题是否可以在因子$\approx 1$的范围内被计算高效地近似。 在\emph{受控松动}(CLuP-SK) 算法成功应用于寻找紧密相关的夏里顿-基尔帕特里克 (SK) 模型的接近基态能量之后，我们在这里提出了 CLuP$\pm$Hop 对应物用于$\pm$Hop 模型。完全提升的随机对偶理论 (fl RDT) [78] 被用于表征 CLuP$\pm$Hop\emph{典型的}动力学。观察到实际性能与理论预测之间有很好的一致性。 特别是，对于$n$小至几千 CLuP$\pm$时，可以达到$\sim 1.77$和$\sim 0.33$作为正负 Hopfield 模型的基态自由能。 同时我们在提升的第6层（6-spl RDT）得到相应的理论热力学极限 ($n\rightarrow\infty$)$\approx 1.7784$和$\approx 0.3281$。 这将霍普菲尔德模型定位在接近基态能量的\emph{通常情况下}易问题。 此外，同样的第六提升层次评估可以揭示两个模型之间的基本内在差异：$+$霍普的近似最优配置彼此为\emph{通常接近}，而$-$霍普的则是\emph{通常相距很远}。 显示英文摘要

摘要： 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}.