v2004.06.19 - Convex Optimization

4.10. LIST RECONSTRUCTION 171minimum eigenvalue. Since the minimum eigenvalue of a symmetric matrixis known to be a concave function (§4.8.4), we calculate its second partialderivative with respect to d 14 evaluated at 2 and find −1/3. We concludethere are no other satisfying values of d 14 . Further, that value of d 14 does notmeet an upper or lower bound of a triangle inequality like (460), so neitherdoes it cause the collapse of any triangle. Because the minimum eigenvalue is0, the affine dimension r of any point list corresponding to D cannot exceedN −2. (§4.7.2)4.10 List reconstructionIsometric reconstruction (§4.5.3) of point list X is generally performed byfactorization of some quantity involving inter-point distance-square data.4.10.1 via GramFor quick reconstruction of a generating list, we may simply performa Cholesky factorization 4.27 of the Gram matrix (350) or (355) fromgiven D ∈ EDM N ; id est, G = X T X where reconstruction X providedby the factorization is upper triangular. Alternatively, we may factorize−VN TDV N =Θ T Θ (379) to find upper triangular reconstruction Θ (377).We now consider how rotation/reflection and translation invariance factorinto a reconstruction. Alternatively, the reader may skip ahead to theexamples.4.10.2 x 1 at the origin. V NAt the stage of reconstruction, we have D ∈ EDM N and we wish to find a generatinglist (§2.2.2) for P − α by factoring positive semidefinite −V T N DV N(484) as suggested in §4.9.1.0.2. One way to factor −V T N DV N is viadiagonalization of symmetric matrices; [26, §5.6] [28] (§A.5, §A.3)− V T NDV N ∆ = QΛQ T (486)QΛQ T ≽ 0 ⇔ Λ ≽ 0 (487)4.27 A very stable numerical algorithm (not requiring definiteness) is given in §F.1.1.5.