v2009.01.01 - Convex Optimization

496 CHAPTER 7. PROXIMITY PROBLEMS
7.0.1 Measurement matrix H
Ideally, we want a given matrix of measurements H ∈ R N×N to conform
with the first three Euclidean metric properties (5.2); to belong to the
intersection of the orthant of nonnegative matrices R N×N
+ with the symmetric
hollow subspace S N h (2.2.3.0.1). Geometrically, we want H to belong to the
polyhedral cone (2.12.1.0.1)
K ∆ = S N h ∩ R N×N
+ (1206)
Yet in practice, H can possess significant measurement uncertainty (noise).
Sometimes realization of an optimization problem demands that its
input, the given matrix H , possess some particular characteristics; perhaps
symmetry and hollowness or nonnegativity. When that H given does not
possess the desired properties, then we must impose them upon H prior to
optimization:
When measurement matrix H is not symmetric or hollow, taking its
symmetric hollow part is equivalent to orthogonal projection on the
symmetric hollow subspace S N h .
When measurements of distance in H are negative, zeroing negative
entries effects unique minimum-distance projection on the orthant of
nonnegative matrices R N×N
+ in isomorphic R N2 (E.9.2.2.3).
7.0.1.1 Order of imposition
Since convex cone K (1206) is the intersection of an orthant with a subspace,
we want to project on that subset of the orthant belonging to the subspace;
on the nonnegative orthant in the symmetric hollow subspace that is, in fact,
the intersection. For that reason alone, unique minimum-distance projection
of H on K (that member of K closest to H in isomorphic R N2 in the Euclidean
sense) can be attained by first taking its symmetric hollow part, and only
then clipping negative entries of the result to 0 ; id est, there is only one
correct order of projection, in general, on an orthant intersecting a subspace: