v2007.11.26 - Convex Optimization

416 CHAPTER 6. CONE OF DISTANCE MATRICES6.5 EDM definition in 11 TAny EDM D corresponding to affine dimension r has representationD(V X ) ∆ = δ(V X V T X )1 T + 1δ(V X V T X ) T − 2V X V T X ∈ EDM N (1020)where R(V X ∈ R N×r )⊆ N(1 T ) = 1 ⊥ ,VX T V X = δ 2 (VX T V X ) and V X is full-rank with orthogonal columns.(1021)Equation (1020) is simply the standard EDM definition (734) with acentered list X as in (816); Gram matrix X T X has been replaced withthe subcompact singular value decomposition (A.6.2) 6.4V X V T X ≡ V T X T XV ∈ S N c ∩ S N + (1022)This means: inner product VX TV X is an r×r diagonal matrix Σ of nonzerosingular values.Vector δ(V X VX T ) may me decomposed into complementary parts byprojecting it on orthogonal subspaces 1 ⊥ and R(1) : namely,P 1 ⊥(δ(VX V T X ) ) = V δ(V X V T X ) (1023)P 1(δ(VX V T X ) ) = 1 N 11T δ(V X V T X ) (1024)Of courseδ(V X V T X ) = V δ(V X V T X ) + 1 N 11T δ(V X V T X ) (1025)by (757). Substituting this into EDM definition (1020), we get theHayden, Wells, Liu, & Tarazaga EDM formula [134,2]whereD(V X , y) ∆ = y1 T + 1y T + λ N 11T − 2V X V T X ∈ EDM N (1026)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 )∆(1027)6.4 Subcompact SVD: V X VXT = Q √ Σ √ ΣQ T ≡ V T X T XV . So VX T is not necessarily XV(5.5.1.0.1), although affine dimension r = rank(VX T ) = rank(XV ). (866)