v2007.09.13 - Convex Optimization

542 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTSFor A,B∈ S N whose eigenvalues λ(A), λ(B)∈ R N are respectivelyarranged in nonincreasing order, and for nonincreasingly ordereddiagonalizations A = W A ΥWA T and B = W B ΛW B T [148] [174,2.1]λ(A) T λ(B) = supU∈ R N×NU T U=I(confer (1506)) where optimal U istr(A T U T BU) ≥ tr(A T B) (1501)U ⋆ = W B W TA ∈ R N×N (1498)We can push that upper bound higher using a result inC.4.2.1:|λ(A)| T |λ(B)| = supU∈ C N×NU H U=IRe tr(A T U H BU) (1484)For step function ψ as defined in (1360), optimal U becomesU ⋆ = W B√δ(ψ(δ(Λ)))H√δ(ψ(δ(Υ))) WTA ∈ C N×N (1485)C.3 Orthogonal Procrustes problemGiven matrices A,B∈ R n×N , their product having full singular valuedecomposition (A.6.3)AB T ∆ = UΣQ T ∈ R n×n (1486)then an optimal solution R ⋆ to the orthogonal Procrustes problemminimize ‖A − R T B‖ FR(1487)subject to R T = R −1maximizes tr(A T R T B) over the nonconvex manifold of orthogonal matrices:[149,7.4.8]R ⋆ = QU T ∈ R n×n (1488)A necessary and sufficient condition for optimalityholds whenever R ⋆ is an orthogonal matrix. [113,4]AB T R ⋆ ≽ 0 (1489)