I’m excited that our paper “A Semidefinite Relaxation for Sums of Heterogeneous Quadratic Forms on the Stiefel Manifold” has been published in the SIAM Journal on Matrix Analysis and Applications. https://doi.org/10.1137/23M1545136. We were inspired to work on this problem after it popped up inside the heteroscedastic PCA problem. It’s a fascinating, simple, general problem with connections to PCA, joint diagonalization, and low-rank semidefinite programs. Applying the standard Schur relaxation to this problem gives a trivial (and incorrect) solution, but one minor change makes the relaxation powerful and even tight in many instances. You can find the code for experiments here.
April 19, 2025