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We consider alignment of sparse graphs, which consists in finding a mapping between the nodes of two graphs which preserves most of the edges. Our approach is to compare local structures in the two graphs, matching two nodes if their neighborhoods are close enough: for correlated ErdH{o}s-Renyi random graphs, this problem can be locally rephrased in terms of testing whether a pair of branching trees is drawn from either a product distribution, or a correlated distribution. We design an optimal test for this problem which gives rise to a message-passing algorithm for graph alignment, which provably returns in polynomial time a positive fraction of correctly matched vertices, and a vanishing fraction of mismatches. With an average degree $lambda = O(1)$ in the graphs, and a correlation parameter $s in [0,1]$, this result holds with $lambda s$ large enough, and $1-s$ small enough, completing the recent state-of-the-art diagram. Tighter conditions for determining whether partial graph alignment (or correlation detection in trees) is feasible in polynomial time are given in terms of Kullback-Leibler divergences.
Random graph alignment refers to recovering the underlying vertex correspondence between two random graphs with correlated edges. This can be viewed as an average-case and noisy version of the well-known graph isomorphism problem. For the correlated
Correlation alignment (CORAL), a representative domain adaptation (DA) algorithm, decorrelates and aligns a labelled source domain dataset to an unlabelled target domain dataset to minimize the domain shift such that a classifier can be applied to pr
Unsupervised domain adaptation (UDA) aims to transfer knowledge from a well-labeled source domain to a different but related unlabeled target domain with identical label space. Currently, the main workhorse for solving UDA is domain alignment, which
Spectral algorithms, such as principal component analysis and spectral clustering, typically require careful data transformations to be effective: upon observing a matrix $A$, one may look at the spectrum of $psi(A)$ for a properly chosen $psi$. The
In arXiv:math/0603621 we introduced the notion of a partial translation $C^*$-algebra for a discrete metric space. Here we demonstrate that several important classical $C^*$-algebras and extensions arise naturally by considering partial translation algebras associated with subspaces of trees.