# Another look at the Gardner problem

arXiv: https://arxiv.org/abs/1306.3979 · HTML: https://mlsanity.com/paper/1306.3979/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jun 2013.

## Built on by (most cited first, 24 of 24)

- [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)
- [Injectivity of ReLU networks: perspectives from statistical physics](https://mlsanity.com/paper/2302.14112/) (2023)
- [No Free Prune: Information-Theoretic Barriers to Pruning at Initialization](https://mlsanity.com/paper/2402.01089/) (2024)
- [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)
- [Fixed width treelike neural networks capacity analysis -- generic activations](https://mlsanity.com/paper/2402.05696/) (2024)
- [Injectivity capacity of ReLU gates](https://mlsanity.com/paper/2410.20646/) (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)
- [Binary perceptron computational gap -- a parametric fl RDT view](https://mlsanity.com/paper/2511.01037/) (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)
- [How abundant are good interpolators?](https://mlsanity.com/paper/2606.06469/) (2026)
- [Exact objectives of random linear programs and mean widths of random polyhedrons](https://mlsanity.com/paper/2403.03637/) (2024)
- [Ridge interpolators in correlated factor regression models -- exact risk analysis](https://mlsanity.com/paper/2406.09183/) (2024)
- [Precise analysis of ridge interpolators under heavy correlations -- a Random Duality Theory view](https://mlsanity.com/paper/2406.09199/) (2024)
- [Theoretical limits of descending ℓ0 sparse-regression ML algorithms](https://mlsanity.com/paper/2410.07651/) (2024)
- [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)

In this paper we revisit one of the classical perceptron problems from the neural networks and statistical physics. In citeGar88 Gardner presented a neat statistical physics type of approach for analyzing what is now typically referred to as the Gardner problem. The problem amounts to discovering a statistical behavior of a spherical perceptron. Among various quantities citeGar88 determined the so-called storage capacity of the corresponding neural network and analyzed its deviations as various perceptron parameters change. In a more recent work citeSchTir02,SchTir03 many of the findings of citeGar88 (obtained on the grounds of the statistical mechanics replica approach) were proven to be mathematically correct. In this paper, we take another look at the Gardner problem and provide a simple alternative framework for its analysis. As a result we reprove many of now known facts and rigorously reestablish a few other results.