v2009.01.01 - Convex Optimization

7.3. THIRD PREVALENT PROBLEM: 531
7.3.3 Constrained affine dimension, Problem 3
When one desires affine dimension diminished further below what can be
achieved via cenv(rank)-minimization as in (1303), spectral projection can be
considered a natural means in light of its successful application to projection
on a rank ρ subset of the positive semidefinite cone in7.1.4.
Yet it is wrong here to zero eigenvalues of −V DV or −V GV or a
variant to reduce affine dimension, because that particular method comes
from projection on a positive semidefinite cone (1241); zeroing those
eigenvalues here in Problem 3 would place an elbow in the projection
path (confer Figure 129) thereby producing a result that is necessarily
suboptimal. Problem 3 is instead a projection on the EDM cone whose
associated spectral cone is considerably different. (5.11.2.3) Proper choice
of spectral cone is demanded by diagonalization of that variable argument to
the objective:
7.3.3.1 Cayley-Menger form
We use Cayley-Menger composition of the Euclidean distance matrix to solve
a problem that is the same as Problem 3 (1286): (5.7.3.0.1)
[ ] [ ]∥ minimize
0 1
T 0 1
T ∥∥∥
2
D
∥ −
1 −D 1 −H
[ ]
F
0 1
T
(1304)
subject to rank
≤ ρ + 2
1 −D
D ∈ EDM N
a projection of H on a generally nonconvex subset (when ρ < N −1) of the
Euclidean distance matrix cone boundary rel∂EDM N ; id est, projection
from the EDM cone interior or exterior on a subset of its relative boundary
(6.6, (1084)).
Rank of an optimal solution is intrinsically bounded above and below;
[ ] 0 1
T
2 ≤ rank
1 −D ⋆ ≤ ρ + 2 ≤ N + 1 (1305)
Our proposed strategy ([ for low-rank ]) solution is projection on that subset
0 1
T
of a spectral cone λ
1 −EDM N (5.11.2.3) corresponding to affine