v2009.01.01 - Convex Optimization

5.10. EDM-ENTRY COMPOSITION 413
of any triangle. Because the smallest eigenvalue is 0, affine dimension r of
any point list corresponding to D cannot exceed N −2. (5.7.1.1)
5.10 EDM-entry composition
Laurent [202,2.3] applies results from Schoenberg (1938) [271] to show
certain nonlinear compositions of individual EDM entries yield EDMs;
in particular,
D ∈ EDM N ⇔ [1 − e −αd ij
] ∈ EDM N ∀α > 0
⇔ [e −αd ij
] ∈ E N ∀α > 0
(997)
where D = [d ij ] and E N is the elliptope (981).
5.10.0.0.1 Proof. (Laurent, 2003) [271] (confer [197])
Lemma 2.1. from A Tour d’Horizon ...on Completion Problems. [202]
The following assertions are equivalent: for D=[d ij , i,j=1... N]∈ S N h and
E N the elliptope in S N (5.9.1.0.1),
(i) D ∈ EDM N
(ii) e −αD ∆ = [e −αd ij
] ∈ E N for all α > 0
(iii) 11 T − e −αD ∆ = [1 − e −αd ij
] ∈ EDM N for all α > 0
⋄
1) Equivalence of Lemma 2.1 (i) (ii) is stated in Schoenberg’s Theorem 1
[271, p.527].
2) (ii) ⇒ (iii) can be seen from the statement in the beginning of section 3,
saying that a distance space embeds in L 2 iff some associated matrix
is PSD. We reformulate it:
Let d =(d ij ) i,j=0,1...N be a distance space on N+1 points
(i.e., symmetric hollow matrix of order N+1) and let
p =(p ij ) i,j=1...N be the symmetric matrix of order N related by:
(A) 2p ij = d 0i + d 0j − d ij for i,j = 1... N
or equivalently
(B) d 0i = p ii , d ij = p ii + p jj − 2p ij for i,j = 1... N