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 pairwise rankings, represented as edge flows on an appropriate graph. Our rank learning method exploits the graph Helmholtzian, which is the graph theoretic analogue of the Helmholtz operator or vector Laplacian, in much the same way the graph Laplacian is an analogue of the Laplace operator or scalar Laplacian. We shall study the graph Helmholtzian using combinatorial Hodge theory, which provides a way to unravel ranking information from edge flows. In particular, we show that every edge flow representing pairwise ranking can be resolved into two orthogonal components, a gradient flow that represents the l2-optimal global ranking and a cyclic (divergence-free) flow that measures the inconsistency of the global ranking obtained — if this large, then it indicates that the data does not have a good global ranking. This cyclic flow can be further decomposed orthogonally into a triangular cyclic flow (curl) and a ‘harmonic’ flow that is globally cyclic but locally acyclic; these provides information on whether inconsistency in the ranking data arises locally or globally. When applied to the problem of rank learning, Hodge decomposition sheds light on whether a given dataset may be globally ranked in a meaningful way or if the data is inherently inconsistent and thus could not have any reasonable global ranking; in the latter case it provides information on the nature of the inconsistency. An obvious advantage over the NP-hardness of Kemeny optimization (which is primarily for ordinal ranking data) is that the discrete Hodge decomposition may be easily computed via a linear least squares regression. We also investigated the l1-projection of edge flows, showing that this has a dual given by correlation maximization over bounded divergence-free flows, and the l1-approximate sparse cyclic ranking, showing that this has a dual given by correlation maximization over bounded curl-free flows. We discuss connections with well-known ordinal ranking techniques such as Kemeny optimization and Borda count from social choice theory.