v2010.10.26 - Convex Optimization

6.4. EDM DEFINITION IN 11 T 5096.4.3 Faces of EDM coneLike the positive semidefinite cone, EDM cone faces are EDM cones.6.4.3.0.1 Exercise. Isomorphic faces.Prove that in high cardinality N , any set of EDMs made via (1202) or (1203)with particular affine dimension r is isomorphic with any set admitting thesame affine dimension but made in lower cardinality.6.4.3.1 smallest face that contains an EDMNow suppose we are given a particular EDM D(V Xp )∈ EDM N correspondingto affine dimension r and parametrized by V Xp in (1185). The EDM cone’ssmallest face that contains D(V Xp ) isF ( EDM N ∋ D(V Xp ) )= { D(V X ) | V X ∈ R N×r , rankV X =r , V T X V X = δ2 (V T X V X ), R(V X)⊆ R(V Xp ) }≃ EDM r+1 (1206)which is isomorphic 6.6 with convex cone EDM r+1 , hence of dimensiondim F ( EDM N ∋ D(V Xp ) ) = (r + 1)r/2 (1207)in isomorphic R N(N−1)/2 . Not all dimensions are represented; e.g., the EDMcone has no two-dimensional faces.When cardinality N = 4 and affine dimension r=2 so that R(V Xp ) is anytwo-dimensional subspace of three-dimensional N(1 T ) in R 4 , for example,then the corresponding face of EDM 4 is isometrically isomorphic with: (1203)EDM 3 = {D ∈ EDM 3 | rank(V DV )≤ 2} ≃ F(EDM 4 ∋ D(V Xp )) (1208)Each two-dimensional subspace of N(1 T ) corresponds to anotherthree-dimensional face.Because each and every principal submatrix of an EDM in EDM N(5.14.3) is another EDM [235,4.1], for example, then each principalsubmatrix belongs to a particular face of EDM N .6.6 The fact that the smallest face is isomorphic with another EDM cone (perhaps smallerthan EDM N ) is implicit in [185,2].