v2009.01.01 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 339
This rank constraint on the composite matrix insures rankX ≤ 5 for any
choice of dimensionally compatible matrices W 1 and W 2 . But to solve this
problem by convex iteration, we alternate solution of semidefinite program
([ ] )
minimize tr W1 X
W 1 , W 2 , X X T Z
W 2
subject to A vec X = y
[ ]
W1 X
X T ≽ 0
W 2
(774)
with semidefinite program
minimize
Z
([ ] ⋆ )
W1 X
tr
X T Z
W 2
subject to 0 ≼ Z ≼ I
trZ = 46 + 81 − 5
(775)
(which has an optimal solution that is known in closed form, p.599) until
a rank-5 composite matrix is found. The semidefiniteness constraints imply
that matrices W 1 and W 2 and Z are symmetric.
With 1000 samples {y i } , convergence occurs in two iterations; 700
samples require more than ten, but reconstruction remains perfect.
Reconstruction is independent of pseudorandom sequence parameters; e.g.,
binary sequences succeed with the same efficiency as Gaussian or uniformly
distributed sequences.
4.6.1 rank-constraint midsummary
We find that this direction matrix idea works well and quite independently
of desired rank or affine dimension. This idea of direction matrix is good
principally because of its simplicity: When confronted with a problem
otherwise convex if not for a rank or cardinality constraint, then that
constraint becomes a linear regularization term in the objective.
There exists a common thread through all these Examples; that being,
convex iteration with a direction matrix as normal to a linear regularization
(a generalization of the well-known trace heuristic). But each problem type
(per Example) possesses its own idiosyncrasies that slightly modify how a
rank-constrained optimal solution is actually obtained: The ball packing