# Lifting/lowering Hopfield models ground state energies

arXiv: https://arxiv.org/abs/1306.3975 · HTML: https://mlsanity.com/paper/1306.3975/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jun 2013.

## Built on by (most cited first, 13 of 13)

- [Binary perceptrons capacity via fully lifted random duality theory](https://mlsanity.com/paper/2312.00073/) (2023)
- [Fl RDT based ultimate lowering of the negative spherical perceptron capacity](https://mlsanity.com/paper/2312.16531/) (2023)
- [Rare dense solutions clusters in asymmetric binary perceptrons -- local entropy via fully lifted RDT](https://mlsanity.com/paper/2506.19276/) (2025)
- [Fixed width treelike neural networks capacity analysis -- generic activations](https://mlsanity.com/paper/2402.05696/) (2024)
- [Capacity of the treelike sign perceptrons neural networks with one hidden layer -- RDT based upper bounds](https://mlsanity.com/paper/2312.08244/) (2023)
- [Lifted RDT based capacity analysis of the 1-hidden layer treelike sign perceptrons neural networks](https://mlsanity.com/paper/2312.08257/) (2023)
- [Deep ReLU networks -- injectivity capacity upper bounds](https://mlsanity.com/paper/2412.19677/) (2024)
- [CLuP practically achieves ~ 1.77 positive and ~ 0.33 negative Hopfield model ground state free energy](https://mlsanity.com/paper/2507.22396/) (2025)
- [Capacity of the Hebbian-Hopfield network associative memory](https://mlsanity.com/paper/2403.01907/) (2024)
- [Parametric RDT approach to computational gap of symmetric binary perceptron](https://mlsanity.com/paper/2601.10628/) (2026)
- [Phase transition of descending phase retrieval algorithms](https://mlsanity.com/paper/2506.18275/) (2025)
- [Phase retrieval with rank d measurements -- descending algorithms phase transitions](https://mlsanity.com/paper/2506.18282/) (2025)
- [A large deviation view of stationarized fully lifted blirp interpolation](https://mlsanity.com/paper/2506.19273/) (2025)

## Abstract (arXiv)

In our recent work citeStojnicHopBnds10 we looked at a class of random optimization problems that arise in the forms typically known as Hopfield models. We viewed two scenarios which we termed as the positive Hopfield form and the negative Hopfield form. For both of these scenarios we defined the binary optimization problems whose optimal values essentially emulate what would typically be known as the ground state energy of these models. We then presented a simple mechanisms that can be used to create a set of theoretical rigorous bounds for these energies. In this paper we create a way more powerful set of mechanisms that can substantially improve the simple bounds given in citeStojnicHopBnds10. In fact, the mechanisms we create in this paper are the first set of results that show that convexity type of bounds can be substantially improved in this type of combinatorial problems.