CLuP practically achieves ~ 1.77 positive and ~ 0.33 negative Hopfield model ground state free energy

We study algorithmic aspects of finding n-dimensional positive and negative Hopfield (±Hop) model ground state free energies. This corresponds to classical maximization of random positive/negative semi-definite quadratic forms over binary left ± frac1√n right n vectors. The key algorithmic question is whether these problems can be computationally efficiently approximated within a factor ≈ 1. Following the introduction and success of 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±Hop counterparts for ±Hop models. Fully lifted random duality theory (fl RDT) [78] is utilized to characterize CLuP±Hop typical dynamics. An excellent agreement between practical performance and theoretical predictions is observed. In particular, for n as small as few thousands CLuP±Hop achieve ~ 1.77 and ~ 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 (nrightarrow∞) limits ≈ 1.7784 and ≈ 0.3281. This positions determining Hopfield models near ground state energies as 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 typically close to each other whereas the -Hop's are typically far away.