v2009.01.01 - Convex Optimization

604 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS
C.3.1.1
Translation of extended list
Suppose an optimal rotation matrix R ⋆ ∈ R n×n were derived as before
from matrix B ∈ R n×N , but B is part of a larger list in the columns of
[C B ]∈ R n×M+N where C ∈ R n×M . In that event, we wish to apply
the rotation/reflection and translation to the larger list. The expression
supplanting the approximation in (1601) makes 1 T of compatible dimension;
R ⋆T [C −B11 T 1 N
BV ] + A11 T 1 N
(1602)
id est, C −B11 T 1 N ∈ Rn×M and A11 T 1 N ∈ Rn×M+N .
C.4 Two-sided orthogonal Procrustes
C.4.0.1
Minimization
Given symmetric A,B∈ S N , each having diagonalization (A.5.2)
A ∆ = Q A Λ A Q T A , B ∆ = Q B Λ B Q T B (1603)
where eigenvalues are arranged in their respective diagonal matrix Λ in
nonincreasing order, then an optimal solution [108]
to the two-sided orthogonal Procrustes problem
R ⋆ = Q B Q T A ∈ R N×N (1604)
minimize ‖A − R T BR‖ F
R
= minimize tr ( A T A − 2A T R T BR + B T B )
subject to R T = R −1 R
subject to R T = R −1 (1605)
maximizes tr(A T R T BR) over the nonconvex manifold of orthogonal matrices.
Optimal product R ⋆T BR ⋆ has the eigenvectors of A but the eigenvalues of B .
[138,7.5.1] The optimal value for the objective of minimization is, by (41)
‖Q A Λ A Q T A −R ⋆T Q B Λ B Q T BR ⋆ ‖ F = ‖Q A (Λ A −Λ B )Q T A ‖ F = ‖Λ A −Λ B ‖ F (1606)
while the corresponding trace maximization has optimal value
sup tr(A T R T BR) = tr(A T R ⋆T BR ⋆ ) = tr(Λ A Λ B ) ≥ tr(A T B) (1607)
R T =R −1