Stability of the replica symmetric solution in diluted perceptron learning

We study the role played by 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 α between the number of patterns M and the dimension N of the perceptron (α = M/N), there exists a critical dilution field hc above which the replica symmetric ansatz becomes unstable. (letter)