v2009.01.01 - Convex Optimization

7.2. SECOND PREVALENT PROBLEM: 523
7.2.2.7 Cumulative summary of rank heuristics
We have studied the perturbation method of rank reduction in4.3, as
well as the trace heuristic (convex envelope method7.2.2.1.1) and log det
heuristic in7.2.2.4. There is another good contemporary method called
LMIRank [247] based on alternating projection (E.10) that does not solve
the ball packing problem presented in5.4.2.2.3.
7.2.2.7.1 Example. Rank regularization enforcing affine dimension.
We apply the convex iteration method from4.4.1 to numerically solve an
instance of Problem 2; a method empirically superior to the foregoing convex
envelope and log det heuristics.
Unidimensional scaling, [89] a historically practical application of
multidimensional scaling (5.12), entails solution of an optimization problem
having local minima whose multiplicity varies as the factorial of point-list
cardinality. Geometrically, it means finding a list constrained to lie in one
affine dimension. In terms of point list, the nonconvex problem is: given
nonnegative symmetric matrix H = [h ij ] ∈ S N ∩ R N×N
+ (1262) whose entries
h ij are all known,
minimize
{x i ∈R}
N∑
(|x i − x j | − h ij ) 2 (1213)
i , j=1
called a raw stress problem [46, p.34] which has an implicit constraint on
dimensional embedding of points {x i ∈ R , i=1... N}. This problem has
proven NP-hard; e.g., [64].
As always, we first transform variables to distance-square D ∈ S N h ; so we
begin with convex problem (1264) on page 517
minimize − tr(V (D − 2Y )V )
D , Y
[ ]
dij y ij
subject to
≽ 0 ,
y ij
h 2 ij
Y ∈ S N h
D ∈ EDM N
rankV T N DV N = 1
N ≥ j > i = 1... N −1
(1281)
that becomes equivalent to (1213) by making explicit the constraint on affine
dimension via rank. The iteration is formed by moving the dimensional