v2007.09.13 - Convex Optimization

396 CHAPTER 6. EDM CONEthe graph. The dimensionless EDM subgraph between each sample andits nearest neighbors is completed from available data and included asinput; one such EDM subgraph completion is drawn superimposed upon theneighborhood graph in Figure 96. 6.3 The consequent dimensionless EDMgraph comprising all the subgraphs is incomplete, in general, because theneighbor number is relatively small; incomplete even though it is a supersetof the neighborhood graph. Remaining distances (those not graphed at all)are squared then made variables within the algorithm; it is this variabilitythat admits unfurling.To demonstrate, consider untying the trefoil knot drawn in Figure 97(a).A corresponding Euclidean distance matrix D = [d ij , i,j=1... N]employing only 2 nearest neighbors is banded having the incomplete form⎡D =⎢⎣0 ď 12 ď 13 ? · · · ? ď 1,N−1 ď 1Nď 12 0 ď 23 ď 24... ? ? ď 2Nď 13 ď 23 0 ď 34. . . ? ? ?? ď 24 ď 34 0...... ? ?................... ?? ? ?...... 0 ď N−2,N−1 ď N−2,Nď 1,N−1 ? ? ?... ď N−2,N−1 0 ď N−1,Nď 1N ď 2N ? ? ? ď N−2,N ď N−1,N 0⎤⎥⎦(987)where ďij denotes a given fixed distance-square. The unfurling algorithmcan be expressed as an optimization problem; constrained distance-squaremaximization:maximize 〈−V , D〉Dsubject to 〈D , e i e T j + e j e T i 〉 1 = 2 ďij ∀(i,j)∈ I(988)rank(V DV ) = 2D ∈ EDM N6.3 Local reconstruction of point position from the EDM submatrix corresponding to acomplete dimensionless EDM subgraph is unique to within an isometry (5.6,5.12).