Open problem: Tightness of maximum likelihood semidefinite relaxations
This addresses a theoretical gap in understanding the tightness of SDP relaxations beyond exact recovery regimes, which is incremental but important for optimization and statistical inference communities.
The paper identifies an unexplained phenomenon where semidefinite programming relaxations of maximum likelihood estimators remain tight in noisy data recovery problems, even when exact recovery is impossible, using the generalized Procrustes problem as an example.
We have observed an interesting, yet unexplained, phenomenon: Semidefinite programming (SDP) based relaxations of maximum likelihood estimators (MLE) tend to be tight in recovery problems with noisy data, even when MLE cannot exactly recover the ground truth. Several results establish tightness of SDP based relaxations in the regime where exact recovery from MLE is possible. However, to the best of our knowledge, their tightness is not understood beyond this regime. As an illustrative example, we focus on the generalized Procrustes problem.