Multifractal analysis of perceptron learning with errors

TL;DR

This paper applies multifractal analysis to the structure of perceptron coupling space, revealing insights into learning, storage, and generalization behaviors under error constraints.

Contribution

It introduces a unified multifractal framework to analyze perceptron learning clusters and their spatial distribution, bridging different learning scenarios.

Findings

Clusters of perceptron cells exhibit multifractal properties.

The analysis provides insights into the spatial distribution of cells.

The approach unifies understanding of various learning and generalization tasks.

Abstract

Random input patterns induce a partition of the coupling space of a perceptron into cells labeled by their output sequences. Learning some data with a maximal error rate leads to clusters of neighboring cells. By analyzing the internal structure of these clusters with the formalism of multifractals, we can handle different storage and generalization tasks for lazy students and absent-minded teachers within one unified approach. The results also allow some conclusions on the spatial distribution of cells.