v2007.09.13 - Convex Optimization

400 CHAPTER 6. EDM CONE6.5 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 (990)where R(V X ∈ R N×r )⊆ N(1 T ) = 1 ⊥ ,V T X V X = δ 2 (V T X V X ) and V X is full-rank with orthogonal columns. (991)Equation (990) is simply the standard EDM definition (705) with a centeredlist X as in (787); Gram matrix X T X has been replaced with thesubcompact singular value decomposition (A.6.2) 6.4V X V T X ≡ V T X T XV ∈ S N c ∩ S N + (992)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 ) (993)P 1(δ(VX V T X ) ) = 1 N 11T δ(V X V T X ) (994)Of courseδ(V X V T X ) = V δ(V X V T X ) + 1 N 11T δ(V X V T X ) (995)by (728). Substituting this into EDM definition (990), we get theHayden, Wells, Liu, & Tarazaga EDM formula [133,2]whereD(V X , y) ∆ = y1 T + 1y T + λ N 11T − 2V X V T X ∈ EDM N (996)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 )∆(997)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 ). (837)