  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.
