v2007.09.13 - Convex Optimization

7.2. SECOND PREVALENT PROBLEM: 4657.2.2.7 Cumulative summary of rank heuristicsWe have studied the perturbation method of rank reduction in4.3, aswell as the trace heuristic (convex envelope method7.2.2.1.1) and log detheuristic in7.2.2.4. There is another good contemporary method calledLMIRank [209] based on alternating projection (E.10) that does not solvethe ball packing problem presented in5.4.2.2.3, so it is not evaluated furtherherein. None of these exceed performance of the convex iteration method forconstraining rank developed in4.4:7.2.2.7.1 Example. Rank regularization enforcing affine dimension.We apply the convex iteration method from4.4.1 to numerically solve aninstance of Problem 2; a method empirically superior to the foregoing convexenvelope and log det heuristics.Unidimensional scaling, [70] a historically practical application ofmultidimensional scaling (5.12), entails solution of an optimization problemhaving local minima whose multiplicity varies as the factorial of point-listcardinality. Geometrically, it means finding a list constrained to lie in oneaffine dimension. In terms of point list, the nonconvex problem is: givennonnegative symmetric matrix H = [h ij ] ∈ S N ∩ R N×N+ (1159) whose entriesh ij are all known, (1110)minimize{x i ∈R}N∑(‖x i − x j ‖ − h ij ) 2 (1178)i , j=1called a raw stress problem [39, p.34] which has an implicit constraint ondimensional embedding of points {x i ∈ R , i = 1... N}. This problem hasproven NP-hard; e.g., [52].As always, we first transform variables to distance-square D ∈ S N h ; so webegin with convex problem (1161) on page 459minimize − tr(V (D − 2Y )V )D , Y[ ]dij y ijsubject to≽ 0 ,y ijh 2 ijY ∈ S N hD ∈ EDM NrankV T N DV N = 1j > i = 1... N −1(1179)