v2010.10.26 - Convex Optimization

666 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTSC.4.2Two-sided orthogonal Procrustes via SVDBy making left- and right-side orthogonal matrices independent, we can pushthe upper bound on trace (1727) a little further: Given real matrices A,Beach having full singular value decomposition (A.6.3)A U A Σ A Q T A ∈ R m×n , B U B Σ B Q T B ∈ R m×n (1737)then a well-known optimal solution R ⋆ , S ⋆ to the problemminimize ‖A − SBR‖ FR , Ssubject to R H = R −1(1738)S H = S −1maximizes re tr(A T SBR) : [314] [288] [52] [194] optimal orthogonal matricesS ⋆ = U A U H B ∈ R m×m , R ⋆ = Q B Q H A ∈ R n×n (1739)[sic] are not necessarily unique [202,7.4.13] because the feasible set is notconvex. The optimal value for the objective of minimization is, by (48)‖U A Σ A Q H A − S ⋆ U B Σ B Q H BR ⋆ ‖ F = ‖U A (Σ A − Σ B )Q H A ‖ F = ‖Σ A − Σ B ‖ F (1740)while the corresponding trace maximization has optimal value [43,III.6.12]sup | tr(A T SBR)| = sup re tr(A T SBR) = re tr(A T S ⋆ BR ⋆ ) = tr(ΣA TΣ B ) ≥ tr(AT B)R H =R −1R H =R −1S H =S −1 S H =S −1 (1741)for which it is necessaryA T S ⋆ BR ⋆ ≽ 0 , BR ⋆ A T S ⋆ ≽ 0 (1742)The lower bound on inner product of singular values in (1741) is due tovon Neumann. Equality is attained if U HA U B =I and QH B Q A =I .C.4.2.1Symmetric matricesNow optimizing over the complex manifold of unitary matrices (B.5.1),the upper bound on trace (1727) is thereby raised: Suppose we are givendiagonalizations for (real) symmetric A,B (A.5)A = W A ΥW TA ∈ S n , δ(Υ) ∈ K M (1743)