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

arXiv: https://arxiv.org/abs/2507.22396 · HTML: https://mlsanity.com/paper/2507.22396/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 86 references that are in the map.
- First posted on arXiv 30 Jul 2025.

## Builds on (most cited first, 17 of 17)

- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization](https://mlsanity.com/paper/2203.05093/) (2022)
- [Subdominant Dense Clusters Allow for Simple Learning and High Computational Performance in Neural Networks with Discrete Synapses](https://mlsanity.com/paper/1509.05753/) (2015)
- [Limits of local algorithms over sparse random graphs](https://mlsanity.com/paper/1304.1831/) (2013)
- [Local entropy as a measure for sampling solutions in Constraint Satisfaction Problems](https://mlsanity.com/paper/1511.05634/) (2015)
- [Optimization of the Sherrington-Kirkpatrick Hamiltonian](https://mlsanity.com/paper/1812.10897/) (2018)
- [Fully lifted random duality theory](https://mlsanity.com/paper/2312.00070/) (2023)
- [Lifting/lowering Hopfield models ground state energies](https://mlsanity.com/paper/1306.3975/) (2013)
- [Binary perceptrons capacity via fully lifted random duality theory](https://mlsanity.com/paper/2312.00073/) (2023)
- [Clustering of solutions in the symmetric binary perceptron](https://mlsanity.com/paper/1911.06756/) (2019)
- [Fl RDT based ultimate lowering of the negative spherical perceptron capacity](https://mlsanity.com/paper/2312.16531/) (2023)
- [Tight Lipschitz Hardness for Optimizing Mean Field Spin Glasses](https://mlsanity.com/paper/2110.07847/) (2021)
- [Rare dense solutions clusters in asymmetric binary perceptrons -- local entropy via fully lifted RDT](https://mlsanity.com/paper/2506.19276/) (2025)
- [A CLuP algorithm to practically achieve ~ 0.76 SK--model ground state free energy](https://mlsanity.com/paper/2507.09247/) (2025)
- [Bounding ground state energy of Hopfield models](https://mlsanity.com/paper/1306.3764/) (2013)
- [The algorithmic hardness threshold for continuous random energy models](https://mlsanity.com/paper/1810.05129/) (2018)
- [Capacity of the Hebbian-Hopfield network associative memory](https://mlsanity.com/paper/2403.01907/) (2024)

## Built on by (most cited first, 3 of 3)

- [Binary perceptron computational gap -- a parametric fl RDT view](https://mlsanity.com/paper/2511.01037/) (2025)
- [Parametric RDT approach to computational gap of symmetric binary perceptron](https://mlsanity.com/paper/2601.10628/) (2026)
- [Ultrametric OGP - parametric RDT symmetric binary perceptron connection](https://mlsanity.com/paper/2604.19712/) (2026)

## Abstract (arXiv)

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.