v2009.01.01 - Convex Optimization

464 CHAPTER 6. CONE OF DISTANCE MATRICES
6.5.3 Faces of EDM cone
Like the positive semidefinite cone, EDM cone faces are EDM cones.
6.5.3.0.1 Exercise. Isomorphic faces.
Prove that in high cardinality N , any set of EDMs made via (1110) or (1111)
with particular affine dimension r is isomorphic with any set admitting the
same affine dimension but made in lower cardinality.
6.5.3.1 Smallest face
Now suppose we are given a particular EDM D(V Xp )∈ EDM N corresponding
to affine dimension r and parametrized by V Xp in (1093). The EDM cone’s
smallest face that contains D(V Xp ) is
F ( 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 (1114)
which is isomorphic 6.7 with the convex cone EDM r+1 , hence of dimension
dim F ( EDM N ∋ D(V Xp ) ) = (r + 1)r/2 (1115)
in isomorphic R N(N−1)/2 . Not all dimensions are represented; e.g., the EDM
cone has no two-dimensional faces.
When cardinality N = 4 and affine dimension r=2 so that R(V Xp ) is any
two-dimensional subspace of three-dimensional N(1 T ) in R 4 , for example,
then the corresponding face of EDM 4 is isometrically isomorphic with: (1111)
EDM 3 = {D ∈ EDM 3 | rank(V DV )≤ 2} ≃ F(EDM 4 ∋ D(V Xp )) (1116)
Each two-dimensional subspace of N(1 T ) corresponds to another
three-dimensional face.
Because each and every principal submatrix of an EDM in EDM N
(5.14.3) is another EDM [202,4.1], for example, then each principal
submatrix belongs to a particular face of EDM N .
6.7 The fact that the smallest face is isomorphic with another EDM cone (perhaps smaller
than EDM N ) is implicit in [159,2].