On Sinkhorn's Algorithm and Choice Modeling
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AI 根据所提供的论文材料归类：摘要明确讨论 Luce 公理下的选择与排序模型及其最大似然估计，直接属于离散选择。
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Sinkhorn 算法与选择模型的联系
- 研究问题
- 基于 Luce 选择公理的选择与排序模型，其最大似然估计如何对应矩阵平衡问题，这种联系能否帮助刻画算法收敛速度？
- 方法与思路
- 作者将 Bradley–Terry–Luce 和 Plackett–Luce 等模型的估计转化为指定边际和的矩阵平衡，以加权二部图分析 Sinkhorn 收敛。
- 材料陈述的贡献
- 摘要称，在存在有限缩放矩阵时建立一般非负矩阵的全局线性收敛，并用图的代数连通性刻画速度，进一步给出精确渐近速率。
- 阅读关注点
- 应核对模型与矩阵平衡的等价映射、有限缩放存在条件，以及图连通性和正边际如何影响全局界、渐近速率与计算表现。
- 阅读边界
- 这是 arXiv 预印本，不能确认同行评审或期刊接收。摘要未给出速率公式及退化情形；二部图结构本身不构成社会网络应用证据。
原文摘要
For a broad class of models widely used in practice for choice and ranking data based on Luce's choice axiom, including the Bradley--Terry--Luce and Plackett--Luce models, we show that the associated maximum likelihood estimation problems are equivalent to a classic matrix balancing problem with target row and column sums. This perspective opens doors between two seemingly unrelated research areas, and allows us to unify existing algorithms in the choice modeling literature as special instances or analogs of Sinkhorn's celebrated algorithm for matrix balancing. We draw inspirations from these connections and resolve some open problems on the study of Sinkhorn's algorithm. We establish the global linear convergence of Sinkhorn's algorithm for non-negative matrices whenever finite scaling matrices exist, and characterize its linear convergence rate in terms of the algebraic connectivity of a weighted bipartite graph. We further derive the sharp asymptotic rate of linear convergence, which generalizes a classic result of Knight (2008). To our knowledge, these are the first quantitative linear convergence results for Sinkhorn's algorithm for general non-negative matrices and positive marginals. Our results highlight the importance of connectivity and orthogonality structures in matrix balancing and Sinkhorn's algorithm, which could be of independent interest. More broadly, the connections we establish in this paper between matrix balancing and choice modeling could also help motivate further transmission of ideas and lead to interesting results in both disciplines.