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 \(\omega(\log n)\), 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.

翻译：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，其中包含一组用户的独立但相关的特征，该问题旨在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，即在给定一个带随机权重的二部图时，目标是识别生成给定权重的潜在匹配。我们研究了一个具有多变量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，展示了它们之间的联系。当数据库特征维度为\(\omega(\log n)\)（其中\(n\)为对齐规模）且无单个特征过强时，数据库对齐的性能阈值收敛至植入匹配的阈值。植入匹配和数据库对齐的最大似然算法均采用线性规划形式，我们研究其松弛形式以更好理解不同条件下各种约束的重要性，并给出可达性和逆界。结果表明，松弛算法的近乎精确对齐阈值与最大似然一致，但精确对齐阈值之间存在差距。我们的分析与结果可扩展至一个用户集未被对齐完全覆盖的非平衡情形。