Many maximum likelihood estimation problems are known to be
intractable in the worst case. A common approach is to consider convex
relaxations of the maximum likelihood estimator (MLE), and relaxations
based on semidefinite programming (SDP) are among the most popular. We
will focus our attention on a certain class of graph-based inverse
problems and show a couple of remarkable phenomena.
In some instances of these problems (such as community detection under
the stochastic block model) the solution to the SDP matches the ground
truth parameters (i.e. achieves exact recovery) for information
theoretically optimal regimes. This is established using new
nonasymptotic bounds for the spectral norm of random matrices with
On other instances of these problems (such as angular
synchronization), the MLE itself tends to not coincide with the ground
truth (although maintaining favorable statistical properties).
Remarkably, these relaxations are often still tight (meaning that the
solution of the SDP matches the MLE). For angular synchronization we
can understand this behavior by analyzing the solutions of certain
randomized Grothendieck problems. However, for many other problems,
such as the multireference alignment problem in signal processing,
this remains a fascinating open problem.