title: Phase transitions in semidefinite relaxations.
abstract: Statistical inference problems arising within signal processing, data mining, and machine learning naturally give rise to hard combinatorial optimization problems. These problems become intractable when the dimensionality of the data is large, as is often the case for modern datasets. A popular idea is to construct convex relaxations of these combinatorial problems, which can be solved efficiently for large-scale datasets. Semidefinite programming (SDP) relaxations are among the most powerful methods in this family and are surprisingly well suited for a broad range of problems where data take the form of matrices or graphs. It has been observed several times that when the statistical noise is small enough, SDP relaxations correctly detect the underlying combinatorial structures. In this paper we develop asymptotic predictions for several detection thresholds, as well as for the estimation error above these thresholds. We study some classical SDP relaxations for statistical problems motivated by graph synchronization and community detection in networks. We map these optimization problems to statistical mechanics models with vector spins and use nonrigorous techniques from statistical mechanics to characterize the corresponding phase transitions. Our results clarify the effectiveness of SDP relaxations in solving high-dimensional statistical problems.
authors.affiliation: Marshall School of Business, University of Southern California, Los Angeles, CA 90089;
authors.forename: Adel
authors.initials: A
authors.lastname: Javanmard
authors.affiliation: Department of Statistics, Stanford University, Stanford, CA 94305; Department of Electrical Engineering, Stanford University, Stanford, CA 94305; montanari@stanford.edu.
authors.forename: Andrea
authors.initials: A
authors.lastname: Montanari
authors.affiliation: Dipartimento di Fisica, Università La Sapienza, 00185 Rome, Italy; Istituto Nazionale di Fisica Nucleare, Sezione di Roma1, 00185 Rome, Italy; Rome unit, Nanotec, Consiglio Nazionale delle Ricerche, 00185 Rome, Italy.
authors.forename: Federico
authors.initials: F
authors.lastname: Ricci-Tersenghi
keywords: community detection; phase transitions; semidefinite programming; synchronization