Phase Transitions in Semidefinite Relaxations

TL;DR

This paper analyzes phase transitions in semidefinite programming relaxations for high-dimensional statistical inference problems, providing asymptotic predictions for detection thresholds and estimation errors using statistical mechanics techniques.

Contribution

It develops asymptotic predictions for detection thresholds and estimation errors in SDP relaxations, connecting statistical inference with phase transition analysis via statistical mechanics.

Findings

Identifies detection thresholds for SDP relaxations in graph problems

Characterizes phase transitions in high-dimensional inference tasks

Clarifies when SDP relaxations effectively detect underlying structures

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 non-rigorous techniques from statistical mechanics to characterize the corresponding phase transitions. Our results clarify the effectiveness of SDP relaxations in solving high-dimensional statistical problems.