v2010.10.26 - Convex Optimization

5.5. INVARIANCE 4355.5.1.0.1 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; [352] [163]α = α c Xb c X1 1 N ∈ P ⊆ A (973)where A represents the list’s affine hull. If we were to associate a point-massm 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 sameas the center of mass under the assumption of uniform mass density acrosspoints. [219] The geometric center always lies in the convex hull P of the list;id est, α c ∈ P because b T c 1=1 and b c ≽ 0 . 5.25 Subtracting the geometriccenter from every list member,X − α c 1 T = X − 1 N X11T = X(I − 1 N 11T ) = XV ∈ R n×N (974)where V is the geometric centering matrix (913). So we have (confer (891))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 N5.5.1.1 Gram-form invarianceFollowing from (975) and the linear Gram-form EDM operator (903):(975)D(G) = D(V GV ) = δ(V GV )1 T + 1δ(V GV ) T − 2V GV ∈ EDM N (976)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 )) (977)because, for any u∈ R N , D(u1 T + 1u T )=0. The collection of all suchdoublets forms the nullspace (993) to the operator; the translation-invariantsubspace S N⊥c (2000) of the symmetric matrices S N . This means matrix Gis not unique and can belong to an expanse more broad than a positivesemidefinite cone; id est, G∈ S N + − S N⊥c . So explains Gram matrix sufficiencyin EDM definition (903). 5.265.25 Any b from α = Xb chosen such that b T 1 = 1, more generally, makes an auxiliaryV -matrix. (B.4.5)5.26 A constraint G1=0 would prevent excursion into the translation-invariant subspace(numerical unboundedness).