v2009.01.01 - Convex Optimization

6.5. EDM DEFINITION IN 11 T 459
by (821). Substituting this into EDM definition (1093), we get the
Hayden, Wells, Liu, & Tarazaga EDM formula [159,2]
where
D(V X , y) ∆ = y1 T + 1y T + λ N 11T − 2V X V T X ∈ EDM N (1099)
λ ∆ = 2‖V X ‖ 2 F = 1 T δ(V X V T X )2 and y ∆ = δ(V X V T X ) − λ
2N 1 = V δ(V XV T X )
(1100)
and y=0 if and only if 1 is an eigenvector of EDM D . Scalar λ becomes
an eigenvalue when corresponding eigenvector 1 exists. 6.5
Then the particular dyad sum from (1099)
y1 T + 1y T + λ N 11T ∈ S N⊥
c (1101)
must belong to the orthogonal complement of the geometric center subspace
(p.671), whereas V X V T X ∈ SN c ∩ S N + (1095) belongs to the positive semidefinite
cone in the geometric center subspace.
Proof. We validate eigenvector 1 and eigenvalue λ .
(⇒) Suppose 1 is an eigenvector of EDM D . Then because
it follows
V T X 1 = 0 (1102)
D1 = δ(V X V T X )1T 1 + 1δ(V X V T X )T 1 = N δ(V X V T X ) + ‖V X ‖ 2 F 1
⇒ δ(V X V T X ) ∝ 1 (1103)
For some κ∈ R +
δ(V X V T X ) T 1 = N κ = tr(V T X V X ) = ‖V X ‖ 2 F ⇒ δ(V X V T X ) = 1 N ‖V X ‖ 2 F1 (1104)
so y=0.
(⇐) Now suppose δ(V X VX T)= λ 1 ; id est, y=0. Then
2N
D = λ N 11T − 2V X V T X ∈ EDM N (1105)
1 is an eigenvector with corresponding eigenvalue λ .
6.5 e.g., when X = I in EDM definition (798).