v2010.10.26 - Convex Optimization

6.7. A GEOMETRY OF COMPLETION 525[372,2.2] The physical process is intuitively described as unfurling,unfolding, diffusing, decompacting, or unraveling. In particular instances,the process is a sort of flattening by stretching until taut (but not bycrushing); e.g., unfurling a three-dimensional Euclidean body resembling abillowy national flag reduces that manifold’s affine dimension to r=2.Data input to the proposed process originates from distances betweenneighboring relatively dense samples of a given manifold. Figure 145 realizesa densely sampled neighborhood; called, neighborhood graph. Essentially,the algorithmic process preserves local isometry between nearest neighborsallowing distant neighbors to excurse expansively by “maximizing variance”(Figure 5). The common number of nearest neighbors to each sample isa data-dependent algorithmic parameter whose minimum value connectsthe 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 145. 6.11 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 146a.A corresponding Euclidean distance matrix D = [d ij , i,j=1... N]employing only 2 nearest neighbors is banded having the incomplete form⎡⎤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... ? ? ?D =? ď 24 ď 34 0...... ? ?(1244)................... ?? ? ?...... 0 ď N−2,N−1 ď N−2,N⎢⎣ ď 1,N−1 ? ? ?...⎥ď N−2,N−1 0 ď N−1,N ⎦ď 1N ď 2N ? ? ? ď N−2,N ď N−1,N 06.11 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).