  We study the maximization of sums of heterogeneous quadratic forms over the
Stiefel manifold, a nonconvex problem that arises in several modern signal
processing and machine learning applications such as heteroscedastic
probabilistic principal component analysis (HPPCA). In this work, we derive a
novel semidefinite program (SDP) relaxation and study a few of its theoretical
properties. We prove a global optimality certificate for the original nonconvex
problem via a dual certificate, which leads to a simple feasibility problem to
certify global optimality of a candidate local solution on the Stiefel
manifold. In addition, our relaxation reduces to an assignment linear program
for jointly diagonalizable problems and is therefore known to be tight in that
case. We generalize this result to show that it is also tight for close-to
jointly diagonalizable problems, and we show that the HPPCA problem has this
characteristic. Numerical results validate our global optimality certificate
and sufficient conditions for when the SDP is tight in various problem
settings.
