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This paper studies the problem of maximizing the sum of traces of matrixquadratic forms on a product of Stiefel manifolds. This orthogonal trace-summaximization (OTSM) problem generalizes many interesting problems such asgeneralized canonical correlation analysis (CCA), Procrustes analysis, andcryo-electron microscopy of the Nobel prize fame. For these applicationsfinding global solutions is highly desirable but it has been unclear how tofind even a stationary point, let alone testing its global optimality. Througha close inspection of Ky Fan's classical result (1949) on the variationalformulation of the sum of largest eigenvalues of a symmetric matrix, and asemidefinite programming (SDP) relaxation of the latter, we first provide asimple method to certify global optimality of a given stationary point of OTSM.This method only requires testing whether a symmetric matrix is positivesemidefinite. A by-product of this analysis is an unexpected strong dualitybetween Shapiro-Botha (1988) and Zhang-Singer (2017). After showing that apopular algorithm for generalized CCA and Procrustes analysis may generateoscillating iterates, we propose a simple fix that provably guaranteesconvergence to a stationary point. The combination of our algorithm andcertificate reveals novel global optima of various instances of OTSM.

Orthogonal Trace-Sum Maximization: Applications, Local Algorithms, and Global Optimality