v2009.01.01 - Convex Optimization

5.5. INVARIANCE 383
5.5.1.0.1 Example. Translating geometric center to origin.
We might choose to shift the geometric center α c of an N-point list {x l }
(arranged columnar in X) to the origin; [305] [137]
α = α ∆ c = Xb ∆ c = 1 X1 ∈ P ⊆ A (878)
N
where A represents the list’s affine hull. If we were to associate a point-mass
m l with each of the points x l in the list, then their center of mass
(or gravity) would be ( ∑ x l m l )/ ∑ m l . The geometric center is the same
as the center of mass under the assumption of uniform mass density across
points. [190] The geometric center always lies in the convex hull P of the list;
id est, α c ∈ P because bc T 1=1 and b c ≽ 0 . 5.22 Subtracting the geometric
center from every list member,
X − α c 1 T = X − 1 N X11T = X(I − 1 N 11T ) = XV ∈ R n×N (879)
where V is the geometric centering matrix (821). So we have (confer (798))
D(X) = D(XV ) = δ(V T X T XV )1 T + 1δ(V T X T XV ) T − 2V T X T XV ∈ EDM N
5.5.1.1 Gram-form invariance
Following from (880) and the linear Gram-form EDM operator (810):
(880)
D(G) = D(V GV ) = δ(V GV )1 T + 1δ(V GV ) T − 2V GV ∈ EDM N (881)
The Gram-form consequently exhibits invariance to translation by a doublet
(B.2) u1 T + 1u T ;
D(G) = D(G − (u1 T + 1u T )) (882)
because, for any u∈ R N , D(u1 T + 1u T )=0. The collection of all such
doublets forms the nullspace (898) to the operator; the translation-invariant
subspace S N⊥
c (1876) of the symmetric matrices S N . This means matrix G
can belong to an expanse more broad than a positive semidefinite cone;
id est, G∈ S N + − S N⊥
c . So explains Gram matrix sufficiency in EDM
definition (810). 5.23
5.22 Any b from α = Xb chosen such that b T 1 = 1, more generally, makes an auxiliary
V -matrix. (B.4.5)
5.23 A constraint G1=0 would prevent excursion into the translation-invariant subspace
(numerical unboundedness).