v2007.11.26 - Convex Optimization

454 CHAPTER 7. PROXIMITY PROBLEMS7.0.1 Measurement matrix HIdeally, we want a given matrix of measurements H ∈ R N×N to conformwith the first three Euclidean metric properties (5.2); to belong to theintersection of the orthant of nonnegative matrices R N×N+ with the symmetrichollow subspace S N h (2.2.3.0.1). Geometrically, we want H to belong to thepolyhedral cone (2.12.1.0.1)K ∆ = S N h ∩ R N×N+ (1133)Yet in practice, H can possess significant measurement uncertainty (noise).Sometimes realization of an optimization problem demands that itsinput, the given matrix H , possess some particular characteristics; perhapssymmetry and hollowness or nonnegativity. When that H given does notpossess the desired properties, then we must impose them upon H prior tooptimization:When measurement matrix H is not symmetric or hollow, taking itssymmetric hollow part is equivalent to orthogonal projection on thesymmetric hollow subspace S N h .When measurements of distance in H are negative, zeroing negativeentries effects unique minimum-distance projection on the orthant ofnonnegative matrices R N×N+ in isomorphic R N2 (E.9.2.2.3).7.0.1.1 Order of impositionSince convex cone K (1133) 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 projectionof H on K (that member of K closest to H in isomorphic R N2 in the Euclideansense) can be attained by first taking its symmetric hollow part, and onlythen clipping negative entries of the result to 0 ; id est, there is only onecorrect order of projection, in general, on an orthant intersecting a subspace:project on the subspace, then project the result on the orthant in thatsubspace. (conferE.9.5)