v2010.10.26 - Convex Optimization

6.7. A GEOMETRY OF COMPLETION 527where ďij denotes a given fixed distance-square. The unfurling algorithm canbe expressed as an optimization problem; constrained total distance-squaremaximization:maximize 〈−V , D〉Dsubject to 〈D , e i e T j + e j e T i 〉 1 = 2 ďij ∀(i,j)∈ I(1245)rank(V DV ) = 2D ∈ EDM Nwhere e i ∈ R N is the i th member of the standard basis, where set I indexesthe given distance-square data like that in (1244), where V ∈ R N×N is thegeometric centering matrix (B.4.1), and where〈−V , D〉 = tr(−V DV ) = 2 trG = 1 ∑d ij (915)Nwhere G is the Gram matrix producing D assuming G1 = 0.If the (rank) constraint on affine dimension is ignored, then problem(1245) becomes convex, a corresponding solution D ⋆ can be found, and anearest rank-2 solution is then had by ordered eigenvalue decompositionof −V D ⋆ V followed by spectral projection (7.1.3) oni,j[R2+0]⊂ R N . Thistwo-step process is necessarily suboptimal. Yet because the decompositionfor the trefoil knot reveals only two dominant eigenvalues, the spectralprojection is nearly benign. Such a reconstruction of point position (5.12)utilizing 4 nearest neighbors is drawn in Figure 146b; a low-dimensionalembedding of the trefoil knot.This problem (1245) can, of course, be written equivalently in terms ofGram matrix G , facilitated by (921); videlicet, for Φ ij as in (889)maximize 〈I , G〉G∈S N csubject to 〈G , Φ ij 〉 = ďijrankG = 2G ≽ 0∀(i,j)∈ I(1246)The advantage to converting EDM to Gram is: Gram matrix G is a bridgebetween point list X and EDM D ; constraints on any or all of thesethree variables may now be introduced. (Example 5.4.2.3.5) Confinement