Fractal Set Theoretic Analysis of Performance Losses for Tuning Training Data in Learning Systems

Abstract

This paper focuses on the evaluation of learning performance in intelligent dynamic processes with supervised learning. Learning dynamics are characterized by basins of attraction generated by state transitions in control space (statespace + parameter space). State uncertainty is modelled as a cellular control space, namely the cell space. Learning performance losses are related to nonseparable basins of attractions with fuzzy boundaries and to their erosions under parameter changes. Basins erosions are analyzed as fingering regions which quickly loose their compactness yielding regions of fractional dimensions and degeneracies due to bifurcation phenomena. We therefore claim that “learning” quality of intelligent dynamic processes should be measured by fractal set theoridc methods.

To this end, we generate in this paper learning patterns as convergence maps using the cell to cell mapping concept. We then evaluate predictability of these patterns based on Lyapunov exponents. Performance measures in training are generated based on box counting fractal dimensions and the lose of reliability is detected by bifurcation phenomena. Illustrative results are reported for a collision free intelligent path planner of a planar robot manipulator.