高斯数据库对齐与高斯植入匹配
Gaussian Database Alignment and Gaussian Planted Matching
AI总结：
本文研究高斯特征下的数据库对齐与植入匹配问题，证明高维条件下两者性能阈值收敛，并通过线性规划松弛分析给出可达性与逆界，揭示几乎精确与精确对齐阈值的关系。
摘要：
数据库对齐是图对齐问题的一个变体：给定一对匿名数据库，其中包含一组用户各自独立但相关的特征，问题在于仅基于相关性识别特征之间的对应关系并对齐匿名用户集。这与植入匹配密切相关，在植入匹配中，给定一个具有随机权重的二分图，目标是识别生成给定权重的底层匹配。我们研究了一个具有多元高斯特征的数据库对齐问题实例，并推导出同时适用于数据库对齐和植入匹配的结果，展示了两者之间的联系。当数据库特征的维度为 \\(ω(\log n)\\)（其中 \\(n\\) 是对齐的规模）且没有任何单个特征过强时，数据库对齐的性能阈值收敛到植入匹配的性能阈值。植入匹配和数据库对齐的最大似然算法都采用线性规划的形式，我们研究了松弛形式，以更好地理解各种约束在不同条件下的重要性，并给出了可达性界和逆界。我们的结果表明，松弛算法的几乎精确对齐阈值与最大似然的阈值一致，而精确对齐阈值之间存在差距。我们的分析和结果扩展到不平衡情形，即一个用户集未被对齐完全覆盖。
英文摘要：
Database alignment is a variant of the graph alignment problem: Given a pair of anonymized databases containing separate yet correlated features for a set of users, the problem is to identify the correspondence between the features and align the anonymized user sets based on correlation alone. This closely relates to planted matching, where given a bigraph with random weights, the goal is to identify the underlying matching that generated the given weights. We study an instance of the database alignment problem with multivariate Gaussian features and derive results that apply both for database alignment and for planted matching, demonstrating the connection between them. The performance thresholds for database alignment converge to that for planted matching when the dimensionality of the database features is \(ω(\log n)\), where \(n\) is the size of the alignment, and no individual feature is too strong. The maximum likelihood algorithms for both planted matching and database alignment take the form of a linear program and we study relaxations to better understand the significance of various constraints under various conditions and present achievability and converse bounds. Our results show that the almost-exact alignment threshold for the relaxed algorithms coincide with that of maximum likelihood, while there is a gap between the exact alignment thresholds. Our analysis and results extend to the unbalanced case where one user set is not fully covered by the alignment.
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