v2009.01.01 - Convex Optimization

502 CHAPTER 7. PROXIMITY PROBLEMS
where we have made explicit an imposed upper bound ρ on affine dimension
r = rankV T NDV N = rankV DV (947)
that is benign when ρ =N−1, and where D = ∆ [d ij ] and ◦√ D = ∆ [ √ d ij ] .
Problems (1215.2) and (1215.3) are Euclidean projections of a vectorized
matrix H on an EDM cone (6.3), whereas problems (1215.1) and (1215.4)
are Euclidean projections of a vectorized matrix −VHV on a PSD cone. 7.2
Problem (1215.4) is not posed in the literature because it has limited
theoretical foundation. 7.3
Analytical solution to (1215.1) is known in closed form for any bound ρ
although, as the problem is stated, it is a convex optimization only in the case
ρ =N−1. We show in7.1.4 how (1215.1) becomes a convex optimization
problem for any ρ when transformed to the spectral domain. When expressed
as a function of point list in a matrix X as in (1213), problem (1215.2)
is a variant of what is known in statistics literature as the stress problem.
[46, p.34] [87] [307] Problems (1215.2) and (1215.3) are convex optimization
problems in D for the case ρ =N−1 , wherein (1215.3) becomes equivalent
to (1214). Even with the rank constraint removed from (1215.2), we will see
that the convex problem remaining inherently minimizes affine dimension.
Generally speaking, each problem in (1215) produces a different result
because there is no isometry relating them. Of the various auxiliary
V -matrices (B.4), the geometric centering matrix V (821) appears in the
literature most often although V N (804) is the auxiliary matrix naturally
consequent to Schoenberg’s seminal exposition [270]. Substitution of any
auxiliary matrix or its pseudoinverse into these problems produces another
valid problem.
Substitution of VN
T for left-hand V in (1215.1), in particular, produces a
different result because
minimize ‖−V (D − H)V ‖ 2 F
D
(1216)
subject to D ∈ EDM N
7.2 Because −VHV is orthogonal projection of −H on the geometric center subspace S N c
(E.7.2.0.2), problems (1215.1) and (1215.4) may be interpreted as oblique (nonminimum
distance) projections of −H on a positive semidefinite cone.
7.3 D ∈ EDM N ⇒ ◦√ D ∈ EDM N , −V ◦√ DV ∈ S N + (5.10)