Another look at the Gardner problem

Abstract

In this paper we revisit one of the classical perceptron problems from the neural networks and statistical physics. In \cite{Gar88} 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 \cite{Gar88} 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 \cite{SchTir02,SchTir03} many of the findings of \cite{Gar88} (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.

arXiv admin note: substantial text overlap with arXiv:1306.3809

References in corpus (6)

• Multitasking associative networks
• Parallel processing in immune networks
• Meshes that trap random subspaces
• Regularly random duality
• Analogue neural networks on correlated random graphs
• Bounding ground state energy of Hopfield models

Cited by in corpus (2)

• Gardner formula for Ising perceptron models at small densities
• Negative spherical perceptron