v2007.09.13 - Convex Optimization

6.10. POSTSCRIPT 435When D is an EDM [189,2]N(D) ⊂ N(1 T ) = {z | 1 T z = 0} (1100)Because [112,2] (E.0.1)thenDD † 1 = 11 T D † D = 1 T (1101)R(1) ⊂ R(D) (1102)6.10 postscriptWe provided an equality (1070) relating the convex cone of Euclidean distancematrices to the convex cone of positive semidefinite matrices. Projection onthe positive semidefinite cone constrained by an upper bound on rank iseasy and well known; [84] simply a matter of truncating a list of eigenvalues.Projection on the positive semidefinite cone with such a rank constraint is,in fact, a convex optimization problem. (7.1.4)Yet, it remains difficult to project on the EDM cone under a constrainton rank or affine dimension. One thing we can do is invoke the Schoenbergcriterion (724) and then project on a positive semidefinite cone under aconstraint bounding affine dimension from above.It is our hope that the equality herein relating EDM and PSD cones willbecome a step toward understanding projection on the EDM cone under arank constraint.