Consistency constraints for overlapping data clustering
  We examine overlapping clustering schemes with functorial constraints, in the spirit of Carlsson--Memoli. This avoids issues arising from the chaining required by partition-based methods. Our principal result shows that any clustering functor is naturally constrained to refine single-linkage clusters and be refined by maximal-linkage clusters. We work in the context of metric spaces with non-expansive maps, which is appropriate for modeling data processing which does not increase information content. 
Jared Culbertson, Dan P. Guralnik, Jakob Hansen, Peter F. Stiller
cs.LG stat.ML