统计排序与组合Hodge理论
Statistical ranking and combinatorial Hodge theory
摘要：
我们提出了一些技术来获得一个全球排名的数据，可能是不完整和不平衡的特征几乎普遍的现代数据集来自电子商务和互联网应用程序。我们主要对基于分数或评级的基数数据感兴趣。从原始的排名数据，我们构建成对的排名，表示为一个适当的图上的边流。我们的统计排名方法使用图亥姆霍兹算子，亥姆霍兹算子或向量拉普拉斯算子的图论模拟，以同样的方式，图拉普拉斯算子是拉普拉斯算子或标量拉普拉斯算子的模拟。我们研究了图亥姆霍兹组合霍奇理论：我们表明，每一个边流表示成对排名可以分解成两个正交分量，梯度流表示L2最优的全球排名和无发散流（循环），衡量全球排名的有效性获得-如果这是大的，那么数据没有一个有意义的全球排名。这种无发散流可以进一步正交分解为旋度流（局部循环）和调和流（局部非循环但全局循环）;这些提供了关于局部或全局是否出现不一致的信息。相对于NP难Kemeny优化的一个明显优点是离散Hodge分解可以通过线性最小二乘回归来计算。我们还研究了边缘流的L1-投影，表明这是对偶的相关最大化有界无发散流，和L1-近似稀疏循环排名，表明这是对偶的相关最大化有界无卷曲流。从社会选择理论和统计学的角度讨论了Kemeny最优化、Borda计数和Kendall-Smith一致性指数之间的关系。
英文摘要：
We propose a number of techniques for obtaining a global ranking from data that may be incomplete and imbalanced -- characteristics almost universal to modern datasets coming from e-commerce and internet applications. We are primarily interested in score or rating-based cardinal data. From raw ranking data, we construct pairwise rankings, represented as edge flows on an appropriate graph. Our statistical ranking method uses the graph Helmholtzian, 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 study the graph Helmholtzian using combinatorial Hodge theory: 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 divergence-free flow (cyclic) that measures the validity of the global ranking obtained -- if this is large, then the data does not have a meaningful 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 arises locally or globally. An obvious advantage over the NP-hard Kemeny optimization is that discrete Hodge decomposition may be computed via a linear least squares regression. We also investigated the L1-projection of edge flows, showing that this is dual to correlation maximization over bounded divergence-free flows, and the L1-approximate sparse cyclic ranking, showing that this is dual to correlation maximization over bounded curl-free flows. We discuss relations with Kemeny optimization, Borda count, and Kendall-Smith consistency index from social choice theory and statistics.
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