v2007.09.13 - Convex Optimization

C.3. ORTHOGONAL PROCRUSTES PROBLEM 543Solution to problem (1487) can reveal rotation/reflection (5.5.2,B.5)of one list in the columns of A with respect to another list B . Solutionis unique if rankBV N = n . [77,2.4.1] The optimal value for objective ofminimization istr ( A T A + B T B − 2AB T R ⋆) = tr(A T A) + tr(B T B) − 2 tr(UΣU T )= ‖A‖ 2 F + ‖B‖2 F − 2δ(Σ)T 1while the optimal value for corresponding trace maximization is(1490)sup tr(A T R T B) = tr(A T R ⋆T B) = δ(Σ) T 1 ≥ tr(A T B) (1491)R T =R −1The same optimal solution R ⋆ solvesC.3.1Effect of translationmaximize ‖A + R T B‖ FR(1492)subject to R T = R −1Consider the impact of dc offset in known lists A,B∈ R n×N on problem(1487). 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(1493)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 (1494)R ⋆ = QU T ∈ R n×n (1488)The desired result is an optimally rotated offset listR ⋆T BV + A(I − V ) ≈ A (1495)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 (1495) reduces to R ⋆T B ≈ A .