Stability of the replica symmetric solution in diluted perceptron learning

TL;DR

This paper investigates how dilution affects the stability of the replica symmetric solution in a perceptron model, identifying a critical dilution field where the solution becomes unstable, with implications for learning and generalization.

Contribution

It provides a stability analysis of the replica symmetric solution in diluted perceptrons, revealing the existence of a critical dilution field for stability transition.

Findings

Existence of a critical dilution field $h_c$ for stability

Replica symmetric solution becomes unstable above $h_c$

Stability depends on the ratio of patterns to perceptron dimension

Abstract

We study the role played by the dilution in the average behavior of a perceptron model with continuous coupling with the replica method. We analyze the stability of the replica symmetric solution as a function of the dilution field for the generalization and memorization problems. Thanks to a Gardner like stability analysis we show that at any fixed ratio $\alpha$ between the number of patterns M and the dimension N of the perceptron ($\alpha=M/N$), there exists a critical dilution field $h_c$ above which the replica symmetric ansatz becomes unstable.