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Multifractality and percolation in the coupling space of perceptrons

M. Weigt·A. Engel
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Abstract

The coupling space of perceptrons with continuous as well as with binary weights gets partitioned into a disordered multifractal by a set of $p=γN$ random input patterns. The multifractal spectrum $f(α)$ can be calculated analytically using the replica formalism. The storage capacity and the generalization behaviour of the perceptron are shown to be related to properties of $f(α)$ which are correctly described within the replica symmetric ansatz. Replica symmetry breaking is interpreted geometrically as a transition from percolating to non-percolating 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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BibTeXRIS

M. Weigt, A. Engel. 1997-01-13. Multifractality and percolation in the coupling space of perceptrons. https://doi.org/10.1103/physreve.55.4552

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