Multifractality and percolation in the coupling space of perceptrons

Details Multifractality and percolation in the coupling space of perceptrons Multifractality and percolation in the coupling space of perceptrons

1996-08-21

arxiv.org/abs/cond-mat/9608092v2

Phys. Rev. E55, 4552 (1997)

Physics

Condensed Matter

Disordered Systems and Neural Networks

13 pages, revtex, 4 eps figures, version accepted for publication in Phys. Rev. E

Scientific paper

10.1103/PhysRevE.55.4552

The coupling space of perceptrons with continuous as well as with binary weights gets partitioned into a disordered multifractal by a set of $p=\gamma N$ random input patterns. The multifractal spectrum $f(\alpha)$ 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 $f(\alpha)$ 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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