v2007.09.13 - Convex Optimization

398 CHAPTER 6. EDM CONEwhere e i ∈ R N is the i th member of the standard basis, where set I indexesthe given distance-square data like that in (987), where V ∈ R N×N is thegeometric centering matrix (B.4.1), and where〈−V , D〉 = tr(−V DV ) = 2 trG = 1 N∑d ij (730)i,jwhere G is the Gram matrix producing D assuming G1 = 0.If we ignore the (rank) constraint on affine dimension, then problem (988)becomes convex, a corresponding solution D ⋆ can be found, and a nearestrank-2 solution can be found by ordered eigen [ decomposition ] of −V D ⋆ VR2followed by spectral projection (7.1.3) on ⊂ R N . This two-step0process is necessarily suboptimal. Yet because the decomposition for thetrefoil knot reveals only two dominant eigenvalues, the spectral projectionis nearly benign. Such a reconstruction of point position (5.12) utilizing4 nearest neighbors is drawn in Figure 97(b); a low-dimensional embeddingof the trefoil knot.This problem (988) can, of course, be written equivalently in terms ofGram matrix G , facilitated by (736); videlicet, for Φ ij as in (703)maximize 〈I , G〉G∈S N csubject to 〈G , Φ ij 〉 = ďijrankG = 2G ≽ 0∀(i,j)∈ I(989)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 5.4.2.2.4) Confining Gto the geometric center subspace suffers no loss of generality and serves notheoretical purpose; numerically, this implicit constraint G1 = 0 keeps Gindependent of its translation-invariant subspace S N⊥c (5.5.1.1, Figure 104)so as not to become unbounded.