A cost function for similarity-based hierarchical clustering

The development of algorithms for hierarchical clustering has been hampered by a shortage of precise objective functions. To help address this situation, we introduce a simple cost function on hierarchies over a set of points, given pairwise similarities between those points. We show that this criterion behaves sensibly in canonical instances and that it admits a top-down construction procedure with a provably good approximation ratio

Homotopy classification of gerbes

Gerbes are locally connected presheaves of groupoids on a small Grothendieck site C. They are classified up to local weak equivalence by path components of a cocycle category taking values in the big 2-groupoid Iso(Gr(C)) consisting of all sheaves of groups on C, their isomorphisms and homotopies. If F is a full sub- presheaf of Iso(Gr(C)) then the set [*,BF] of morphisms in the homotopy category of simplicial presheaves classifies gerbes locally weakly equivalent to objects of F. Id St(пF)is the stack completion of the fundamental groupoid(пF)of F if L is a global section of St(пF) and if FL is the homotopy fibre over L of the canonical map BF --> B St(пF), then [*FL] is in bijective correspondence with Giraud's non-abelian cohomology object H2 (C, L) of equivalence classes of gerbes with band L