Linear embedding of binary hierarchies and its applications

Abstract

The discrete binary hierarchy (DBH) is a concept underlyingmany important issues in analysis of complex systems: knowledge structures, testand-search organization, evolutionary trees, taxonomy, data handling, etc. It appears that any DBH corresponds to an orthonormal basis of the Euclidean space related to the hierarchy leaves. The properties of these bases form a mathematical framework which can be applied to such problems as clustering and multiresolution image/signal processing. Clustering applications are based on a DBH-based analogue of the singular-value-decomposition of data matrices. A theoretical support for a method in divisive clustering is provided along with some decomposition-based interpretation aids. Data processing applications appear parallel to those involving the concepts of wavelets and quadtrees. However, DBH-based techniques seem to offer some potential improvements based on relaxing “continuity and homogeneity” restrictions of classical theories.