v2010.10.26 - Convex Optimization

5.10. EDM-ENTRY COMPOSITION 465of any triangle. Because the smallest eigenvalue is 0, affine dimension r ofany point list corresponding to D cannot exceed N −2. (5.7.1.1) 5.10 EDM-entry compositionLaurent [235,2.3] applies results from Schoenberg (1938) [313] to showcertain nonlinear compositions of individual EDM entries yield EDMs;in particular,D ∈ EDM N ⇔ [1 − e −αd ij] ∈ EDM N ∀α > 0 (a)⇔ [e −αd ij] ∈ E N ∀α > 0 (b)(1092)where D = [d ij ] and E N is the elliptope (1076).5.10.0.0.1 Proof. (Monique Laurent, 2003) [313] (confer [227])Lemma 2.1. from A Tour d’Horizon ... on Completion Problems.[235] For D=[d ij , i,j=1... N]∈ S N h and E N the elliptope in S N(5.9.1.0.1), the following assertions are equivalent:(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[313, 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 matrixis 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 letp =(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... Nor equivalently(B) d 0i = p ii , d ij = p ii + p jj − 2p ij for i,j = 1... N