Abstract
The coupling space of perceptrons with continuous as well as with binary weights gets partitioned into a disordered multifractal by a set of p=\ensuremath{\gamma}N random input patterns. The multifractal spectrum f(\ensuremath{\alpha}) can be calculated analytically using the replica formalism. The storage capacity and the generalization behavior of the perceptron are shown to be related to properties of f(\ensuremath{\alpha}) which are correctly described within the replica symmetric Ansatz. Replica symmetry breaking is interpreted geometrically as a transition from percolating to nonpercolating cells. The existence of empty cells gives rise to singularities in the multifractal spectrum. The analytical results for binary couplings are corroborated by numerical studies.
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