v2010.10.26 - Convex Optimization

522 CHAPTER 6. CONE OF DISTANCE MATRICESWhen the Finsler criterion (1242) is applied despite lower affinedimension, the constant κ can go to infinity making the test −D+κ11 T ≽ 0impractical for numerical computation. Chabrillac & Crouzeix invent acriterion for the semidefinite case, but is no more practical: for D ∈ S N hD ∈ EDM N⇔(1243)∃κ p >0 ∀κ≥κ p , −D − κ11 T [sic] has exactly one negative eigenvalue6.7 A geometry of completionIt is not known how to proceed if one wishes to restrict thedimension of the Euclidean space in which the configuration ofpoints may be constructed.−Michael W. Trosset (2000) [352,1]Given an incomplete noiseless EDM, intriguing is the question of whether alist in X ∈ R n×N (76) may be reconstructed and under what circumstancesreconstruction is unique. [2] [4] [5] [6] [8] [17] [67] [205] [217] [234] [235] [236]If one or more entries of a particular EDM are fixed, then geometricinterpretation of the feasible set of completions is the intersection of the EDMcone EDM N in isomorphic subspace R N(N−1)/2 with as many hyperplanes asthere are fixed symmetric entries. 6.10 Assuming a nonempty intersection,then the number of completions is generally infinite, and those correspondingto particular affine dimension r