v2007.11.26 - Convex Optimization

460 CHAPTER 7. PROXIMITY PROBLEMSwhere we have made explicit an imposed upper bound ρ on affine dimensionr = rankV T NDV N = rankV DV (876)that is benign when ρ =N−1, and where D ∆ = [d ij ] and ◦√ D ∆ = [ √ d ij ] .Problems (1142.2) and (1142.3) are Euclidean projections of a vectorizedmatrix H on an EDM cone (6.3), whereas problems (1142.1) and (1142.4) areEuclidean projections of a vectorized matrix −VHV on a PSD cone. Problem(1142.4) is not posed in the literature because it has limited theoreticalfoundation. 7.2Analytical solution to (1142.1) is known in closed form for any bound ρalthough, as the problem is stated, it is a convex optimization only in the caseρ =N−1. We show in7.1.4 how (1142.1) becomes a convex optimizationproblem for any ρ when transformed to the spectral domain. When expressedas a function of point list in a matrix X as in (1140), problem (1142.2)is a variant of what is known in statistics literature as the stress problem.[39, p.34] [68] [269] Problems (1142.2) and (1142.3) are convex optimizationproblems in D for the case ρ =N −1. Even with the rank constraint removedfrom (1142.2), we will see the convex problem remaining inherently minimizesaffine dimension.Generally speaking, each problem in (1142) produces a different resultbecause there is no isometry relating them. Of the various auxiliaryV -matrices (B.4), the geometric centering matrix V (757) appears in theliterature most often although V N (740) is the auxiliary matrix naturallyconsequent to Schoenberg’s seminal exposition [236]. Substitution of anyauxiliary matrix or its pseudoinverse into these problems produces anothervalid problem.Substitution of VNT for left-hand V in (1142.1), in particular, produces adifferent result becauseminimize ‖−V (D − H)V ‖ 2 FD(1143)subject to D ∈ EDM N7.2 D ∈ EDM N ⇒ ◦√ D ∈ EDM N , −V ◦√ DV ∈ S N + (5.10)