v2010.10.26 - Convex Optimization

6.4. EDM DEFINITION IN 11 T 503That is easily proven false by counterexample via (1183), for then( ◦√ D 1 + ◦√ D 2 ) ◦( ◦√ D 1 + ◦√ D 2 ) would need to be a member of EDM N .Notwithstanding, √EDM N ⊆ EDM N (1184)by (1094) (Figure 138), and we learn how to transform a nonconvexproximity problem in the natural coordinates √ d ij to a convex optimizationin7.2.1.6.4 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 (1185)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.(1186)Equation (1185) is simply the standard EDM definition (891) with acentered list X as in (975); Gram matrix X T X has been replaced withthe subcompact singular value decomposition (A.6.2) 6.3V X V T X ≡ V T X T XV ∈ S N c ∩ S N + (1187)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 ) (1188)P 1(δ(VX V T X ) ) = 1 N 11T δ(V X V T X ) (1189)Of courseδ(V X V T X ) = V δ(V X V T X ) + 1 N 11T δ(V X V T X ) (1190)6.3 Subcompact SVD: V X VX T 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 ). (1032)