v2009.01.01 - Convex Optimization

C.3. ORTHOGONAL PROCRUSTES PROBLEM 603
Optimal solution R ⋆ can reveal rotation/reflection (5.5.2,B.5) of one
list in the columns of matrix A with respect to another list in B . Solution
is unique if rankBV N = n . [96,2.4.1] In the case that A is a vector and
permutation of B , solution R ⋆ is not necessarily a permutation matrix
(4.6.0.0.3) although the optimal objective will be zero. More generally,
the optimal value for objective of minimization is
tr ( 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 1
while the optimal value for corresponding trace maximization is
(1596)
sup tr(A T R T B) = tr(A T R ⋆T B) = δ(Σ) T 1 ≥ tr(A T B) (1597)
R T =R −1
The same optimal solution R ⋆ solves
C.3.1
Effect of translation
maximize ‖A + R T B‖ F
R
(1598)
subject to R T = R −1
Consider the impact on problem (1593) of dc offset in known lists
A,B∈ R n×N . Rotation of B there is with respect to the origin, so better
results may be obtained if offset is first accounted. Because the geometric
centers of the lists AV and BV are the origin, instead we solve
minimize ‖AV − R T BV ‖ F
R
(1599)
subject to R T = R −1
where V ∈ S N is the geometric centering matrix (B.4.1). Now we define the
full singular value decomposition
and an optimal rotation matrix
AV B T ∆ = UΣQ T ∈ R n×n (1600)
R ⋆ = QU T ∈ R n×n (1594)
The desired result is an optimally rotated offset list
R ⋆T BV + A(I − V ) ≈ A (1601)
which most closely matches the list in A . Equality is attained when the lists
are precisely related by a rotation/reflection and an offset. When R ⋆T B=A
or B1=A1=0, this result (1601) reduces to R ⋆T B ≈ A .