## Graph Helmholtzian and rank learning

author: Lek-Heng Lim, UC Berkeley
published: Dec. 20, 2008,   recorded: December 2008,   views: 197
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# Slides

0:00 Slides Graph Helmholtzian and Rank Learning Modern ranking data Old problems with ranking New problems with ranking Example: Netflix customer-product rating Objective Local inconsistencies Basic model for rank learning Rank aggregation Pairwise rank aggregation More second order statistics Functions on graph Hilbert space of forms Operators Properties Helmholtz decomposition Hodge theory: matrix theoretic Hodge theory: graph theoretic Rank aggregation problem revisited Answer: not always Boundary of a boundary is empty Illustration Harmonic rankings: locally consistent but globally inconsistent - Questions - Questions - Questions - Questions - Questions - Questions

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# Description

The graph Helmholtzian is the graph theoretic analogue of the Helmholtz operator or vector Laplacian, in much the same way the graph Laplacian is the analogue of the Laplace operator or scalar Laplacian. We will see that a decomposition associated with the graph Helmholtzian provides a way to learn ranking information from incomplete, imbalanced, and cardinal score-based data. In this framework, an edge flow representing pairwise ranking is orthogonally resolved into a gradient flow (acyclic) that represents the L2-optimal global ranking and a divergence-free flow (cyclic) that quantifies the inconsistencies. If the latter is large, then the data does not admit a statistically meaningful global ranking. A further decomposition of the inconsistent component into a curl flow (locally cyclic) and a harmonic flow (locally acyclic) provides information on the validity of small- and large-scale comparisons of alternatives. This is joint work with Xiaoye Jiang, Yuan Yao, and Yinyu Ye.