v2010.10.26 - Convex Optimization

7.4. CONCLUSION 589filtering, multidimensional scaling or principal component analysis (5.12),medical imaging (Figure 107), molecular conformation (Figure 127), andsensor-network localization or wireless location (Figure 85).There has been little progress in spectral projection since the discovery byEckart & Young in 1936 [129] leading to a formula for projection on a rank ρsubset of a positive semidefinite cone (2.9.2.1). [146] The only closed-formspectral method presently available for solving proximity problems, having aconstraint on rank, is based on their discovery (Problem 1,7.1,5.13).One popular recourse is intentional misapplication of Eckart & Young’sresult by introducing spectral projection on a positive semidefinite coneinto Problem 3 via D(G) (903), for example. [73] Since Problem 3instead demands spectral projection on the EDM cone, any solutionacquired that way is necessarily suboptimal.A second recourse is problem redesign: A presupposition to allproximity problems in this chapter is that matrix H is given.We considered H having various properties such as nonnegativity,symmetry, hollowness, or lack thereof. It was assumed that if H didnot already belong to the EDM cone, then we wanted an EDM closestto H in some sense; id est, input-matrix H was assumed corruptedsomehow. For practical problems, it withstands reason that such aproximity problem could instead be reformulated so that some or allentries of H were unknown but bounded above and below by knownlimits; the norm objective is thereby eliminated as in the developmentbeginning on page 323. That particular redesign (the art, p.8), in termsof the Gram-matrix bridge between point-list X and EDM D , at onceencompasses proximity and completion problems.A third recourse is to apply the method of convex iteration just likewe did in7.2.2.7.1. This technique is applicable to any semidefiniteproblem requiring a rank constraint; it places a regularization term inthe objective that enforces the rank constraint.