No abstract

We propose a technique that we call HodgeRank for ranking data that may be incomplete and imbalanced, characteristics common in 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 statistical ranking 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 l 2 -optimal global ranking and a divergence-free flow (cyclic) that measures the validity of the global ranking obtained-if this is large, then it indicates that the data does not have a good global ranking. This divergence-free flow can be further decomposed orthogonally into a curl flow (locally cyclic) and a harmonic flow (locally acyclic but globally cyclic); these provides information on whether inconsistency in the ranking data arises locally or globally. When applied to statistical ranking problems, 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 inconsistencies. An obvious advantage over the NP-hardness of Kemeny optimization is that HodgeRank may be easily computed via a linear least squares regression. We also discuss connections with well-known ordinal ranking techniques such as Kemeny optimization and Borda count from social choice theory.

In multi-Higgs-boson extensions of the standard model, tree-level flavor-changing neutral currents exist naturally, unless suppressed by some symmetry. For a given rate, the exchanged scalar or pseudoscalar mass is very sensitive to the flavor-changing coupling between the first two generations. Since the Yukawa couplings of the first two generations are unknown and certainly very small, bounds which rely on some assumed value of this flavor-changing coupling are quite dubious. One might expect the size (and reliability) of the Yukawa couplings involving the third generation to be greater. In this paper, we consider processes involving T'S and B's, and determine the bounds on the flavor-changing couplings which involve third-generation fields. The strongest bound in the quark sector comes from B-B mixing and in the lepton sector, surprisingly, from p-+ey. It is then noted that the flavor-changing couplings in the quark sector are related to those in the lepton sector in many grand unified theories, and one can ask whether an analysis of rare r decays or rare B decays will provide the strongest constraints. We show that rare B decays provide the strongest bounds, and that no useful information can be obtained from rare r decays. It is also noted that the most promising decay modes are B-Kpr and B , +~T , and we urge experimenters to look for rare decay modes of the B in which a r is in the final state.

In this paper, we study an online learning algorithm in Reproducing Kernel Hilbert Spaces (RKHSs) and general Hilbert spaces. We present a general form of the stochastic gradient method to minimize a quadratic potential function by an independent identically distributed (i.i.d.) sample sequence, and show a probabilistic upper bound for its convergence.