# Clustering of solutions in the symmetric binary perceptron

arXiv: https://arxiv.org/abs/1911.06756 · HTML: https://mlsanity.com/paper/1911.06756/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Nov 2019.

## Built on by (most cited first, 12 of 12)

- [Learning through atypical "phase transitions" in overparameterized neural networks](https://mlsanity.com/paper/2110.00683/) (2021)
- [Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron](https://mlsanity.com/paper/2102.13069/) (2021)
- [Rare dense solutions clusters in asymmetric binary perceptrons -- local entropy via fully lifted RDT](https://mlsanity.com/paper/2506.19276/) (2025)
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- [CLuP practically achieves ~ 1.77 positive and ~ 0.33 negative Hopfield model ground state free energy](https://mlsanity.com/paper/2507.22396/) (2025)
- [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)
- [Phase transition of descending phase retrieval algorithms](https://mlsanity.com/paper/2506.18275/) (2025)
- [Optimal spectral initializers impact on phase retrieval phase transitions -- an RDT view](https://mlsanity.com/paper/2506.18279/) (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)
- [Ultrametric OGP - parametric RDT symmetric binary perceptron connection](https://mlsanity.com/paper/2604.19712/) (2026)

## Abstract (arXiv)

The geometrical features of the (non-convex) loss landscape of neural network models are crucial in ensuring successful optimization and, most importantly, the capability to generalize well. While minimizers' flatness consistently correlates with good generalization, there has been little rigorous work in exploring the condition of existence of such minimizers, even in toy models. Here we consider a simple neural network model, the symmetric perceptron, with binary weights. Phrasing the learning problem as a constraint satisfaction problem, the analogous of a flat minimizer becomes a large and dense cluster of solutions, while the narrowest minimizers are isolated solutions. We perform the first steps toward the rigorous proof of the existence of a dense cluster in certain regimes of the parameters, by computing the first and second moment upper bounds for the existence of pairs of arbitrarily close solutions. Moreover, we present a non rigorous derivation of the same bounds for sets of y solutions at fixed pairwise distances.