v2010.10.26 - Convex Optimization

552 CHAPTER 7. PROXIMITY PROBLEMSThis is best understood referring to Figure 152: Suppose nonnegativeinput H is demanded, and then the problem realization correctly projectsits input first on S N h and then directly on C = EDM N . That demandfor nonnegativity effectively requires imposition of K on input H prior tooptimization so as to obtain correct order of projection (on S N h first). Yetsuch an imposition prior to projection on EDM N generally introduces anelbow into the path of projection (illustrated in Figure 153) caused bythe technique itself; that being, a particular proximity problem realizationrequiring nonnegative input.Any procedure for imposition of nonnegativity on input H can only beincorrect in this circumstance. There is no resolution unless input H isguaranteed nonnegative with no tinkering. Otherwise, we have no choice butto employ a different problem realization; one not demanding nonnegativeinput.7.0.2 Lower boundMost of the problems we encounter in this chapter have the general form:minimize ‖B − A‖ FBsubject to B ∈ C(1306)where A ∈ R m×n is given data. This particular objective denotes Euclideanprojection (E) of vectorized matrix A on the set C which may or may not beconvex. When C is convex, then the projection is unique minimum-distancebecause Frobenius’ norm when squared is a strictly convex function ofvariable B and because the optimal solution is the same regardless of thesquare (513). When C is a subspace, then the direction of projection isorthogonal to C .Denoting by AU A Σ A QA T and BU BΣ B QB T their full singular valuedecompositions (whose singular values are always nonincreasingly ordered(A.6)), there exists a tight lower bound on the objective over the manifoldof orthogonal matrices;‖Σ B − Σ A ‖ F ≤infU A ,U B ,Q A ,Q B‖B − A‖ F (1307)This least lower bound holds more generally for any orthogonally invariantnorm on R m×n (2.2.1) including the Frobenius and spectral norm [328,II.3].[202,7.4.51]