Hodge theory for combinatorial geometries

Hodge Theory for Combinatorial Geometries

The matroid is called loopless if the empty subset of E is closed, and is called a combinatorial geometry if in addition all single element subsets of E are closed. A closed subset of E is called a flat of M, and every subset of E has a well-defined rank and corank in the poset of all flats of M. The notion of matroid played a fundamental role in graph theory, coding theory, combinatorial optim...

Learning to rank with combinatorial Hodge theory

We propose a number of techniques for learning a global ranking from data that may be incomplete and imbalanced — characteristics that are almost universal to modern datasets coming from e-commerce and internet applications. We are primarily interested in cardinal data based on scores or ratings though our methods also give specific insights on ordinal data. From raw ranking data, we construct ...

Statistical ranking and combinatorial Hodge theory

We propose a number of techniques for obtaining a global ranking from data that may be incomplete and imbalanced — characteristics that are almost universal to modern datasets coming from e-commerce and internet applications. We are primarily interested in cardinal data based on scores or ratings though our methods also give specific insights on ordinal data. From raw ranking data, we construct...

Modular Constructions for Combinatorial Geometries

R. Stanley, in an investigation of modular flats in geometries (Algebra Universalis 1-2 (1971), 214—217), proved that the characteristic polynomial xW of a modular flat x divides the characteristic polynomial x(G) of a geometry G. In this paper we identify the quotient: THEOREM. // x is a modular flat of G, x(G)/x(x) = X(7^(G))/(\ 1), where TX{G) is the complete Brown truncation of G by x. (The...

Matching Theorems for Combinatorial Geometries