v2010.10.26 - Convex Optimization

504 CHAPTER 6. CONE OF DISTANCE MATRICESby (913). Substituting this into EDM definition (1185), we get theHayden, Wells, Liu, & Tarazaga EDM formula [185,2]whereD(V X , y) y1 T + 1y T + λ N 11T − 2V X V T X ∈ EDM N (1191)λ 2‖V X ‖ 2 F = 1 T δ(V X V T X )2 and y δ(V X V T X ) − λ2N 1 = V δ(V XV T X )(1192)and y=0 if and only if 1 is an eigenvector of EDM D . Scalar λ becomesan eigenvalue when corresponding eigenvector 1 exists. 6.4Then the particular dyad sum from (1191)y1 T + 1y T + λ N 11T ∈ S N⊥c (1193)must belong to the orthogonal complement of the geometric center subspace(p.731), whereas V X V T X ∈ SN c ∩ S N + (1187) belongs to the positive semidefinitecone in the geometric center subspace.Proof. We validate eigenvector 1 and eigenvalue λ .(⇒) Suppose 1 is an eigenvector of EDM D . Then becauseit followsV T X 1 = 0 (1194)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 (1195)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 (1196)so y=0.(⇐) Now suppose δ(V X VX T)= λ 1 ; id est, y=0. Then2ND = λ N 11T − 2V X V T X ∈ EDM N (1197)1 is an eigenvector with corresponding eigenvalue λ . 6.4 e.g., when X = I in EDM definition (891).