v2006.03.09 - Convex Optimization

488 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTSwhile the optimal value for corresponding trace maximization issupR T =R −1 tr(A T R T B) = tr(A T R ⋆T B) = δ(Σ) T 1 ≥ tr(A T B) (1302)The same optimal solution R ⋆ solvesmaximize ‖A + R T B‖ FR(1303)subject to R T = R −1C.4.1Effect of translationConsider the impact of dc offset in known lists A,B∈ R n×N on problem(1298). Rotation of B there is with respect to the origin, so better resultsmay be obtained if offset is first accounted. Because the geometric centersof the lists AV and BV are the origin, instead we solveminimize ‖AV − R T BV ‖ FR(1304)subject to R T = R −1where V ∈ S N is the geometric centering matrix (B.4.1). Now we define thefull singular value decompositionand an optimal rotation matrixAV B T ∆ = UΣQ T ∈ R n×n (1305)R ⋆ = QU T ∈ R n×n (1299)The desired result is an optimally rotated offset listR ⋆T BV + A(I − V ) ≈ A (1306)which most closely matches the list in A . Equality is attained when the listsare precisely related by a rotation/reflection and an offset. When R ⋆T B=Aor B1=A1=0, this result (1306) reduces to R ⋆T B ≈ A .