v2006.03.09 - Convex Optimization

5.5. EDM DEFINITION IN 11 T 3255.5.3.3 Smallest faceNow suppose we are given a particular EDM D(V Xp )∈ EDM N correspondingto affine dimension r and parametrized by V Xp in (737). 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 ) }which is isomorphic 5.5 with the convex cone EDM r+1 , hence of dimension(756)dim F ( EDM N ∋ D(V Xp ) ) = (r + 1)r/2 (757)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: (748)EDM 3 = {D ∈ EDM 3 | rank(V DV )≤ 2} ≃ F(EDM 4 ∋ D(V Xp )) (758)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(4.14.3) is another EDM [145,4.1], for example, then each principalsubmatrix belongs to a particular face of EDM N .5.5.3.4 Open questionThis result (757) is analogous to that for the positive semidefinite cone,although the question remains open whether all faces of EDM N (whosedimension is less than the dimension of the cone) are exposed like they arefor the positive semidefinite cone. (2.9.2.2) [224]5.5 The fact that the smallest face is isomorphic with another (perhaps smaller) EDMcone is implicit in [110,2].