Metric recovery from directed unweighted graphs
Author(s)Hashimoto, Tatsunori Benjamin; Sun, Yi; Jaakkola, Tommi S
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We analyze directed, unweighted graphs obtained from x[subscript i] ∈ R[superscript d] by connecting vertex i to j iff |x[subscript i] − x[subscript j]| < ε(x[subscript i]). Examples of such graphs include k-nearest neighbor graphs, where ε(x[subscript i]) varies from point to point, and, arguably, many real-world graphs such as copurchasing graphs. We ask whether we can recover the underlying Euclidean metric ε(x[subscript i]) and the associated density p(xi) given only the directed graph and d. We show that consistent recovery is possible up to isometric scaling when the vertex degree is at least ω(n[superscript 2/(2+d)] log(n)[superscript d/(d+2)]). Our estimator is based on a careful characterization of a random walk over the directed graph and the associated continuum limit. As an algorithm, it resembles the PageRank centrality metric. We demonstrate empirically that the estimator performs well on simulated examples as well as on real-world co-purchasing graphs even with a small number of points and degree scaling as low as log(n).
DepartmentMassachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Department of Mathematics
Proceedings of Machine Learning Research
Hashimoto, Tatsunori, Yi Sun, and Tommi S. Jaakkola. "Metric recovery from directed unweighted graphs." Proceedings of the 18th International Conference on Artificial Intelligence and Statistics (AISTATS) 2015, 9-12 May, 2015, San Diego, California, PLMR, 2015. © The Authors
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