Author:
Giovanni Rossi
Affiliation:
University of Bologna, Italy
Keyword(s):
Hamming Distance, Partition Lattice, Hasse Diagram, Weighted Graph, Geodesic Distance, Path.
Related
Ontology
Subjects/Areas/Topics:
Applications
;
Bioinformatics and Systems Biology
;
Biomedical Engineering
;
Biomedical Signal Processing
;
Biometrics
;
Biometrics and Pattern Recognition
;
Classification
;
Clustering
;
Multimedia
;
Multimedia Signal Processing
;
Pattern Recognition
;
Similarity and Distance Learning
;
Software Engineering
;
Telecommunications
;
Theory and Methods
Abstract:
Developing from a concern in bioinformatics, this work analyses alternative metrics between partitions. From
both theoretical and applicative perspectives, a useful and interesting distance between any two partitions is
HD, which counts the number of atoms finer than either one but not both. While faithfully reproducing the
traditional Hamming distance between subsets, HD is very sensible and computable through scalar products
between Boolean vectors. It properly deals with complements and axiomatically resembles the entropy-based
variation of information VI distance. Entire families of metrics (including HD and VI) obtain as minimal
paths in the weighted graph given by the Hasse diagram: submodular weighting functions yield path-based
distances visiting the join (of any two partitions), whereas supermodularity leads to visit the meet. This yields
an exact (rather than heuristic) approach to the consensus partition (combinatorial optimization) problem.