v2009.01.01 - Convex Optimization

454 CHAPTER 6. CONE OF DISTANCE MATRICES
three-dimensional Euclidean body resembling a billowy national flag reduces
that manifold’s affine dimension to r=2.
Data input to the proposed process originates from distances between
neighboring relatively dense samples of a given manifold. Figure 116 realizes
a densely sampled neighborhood; called, neighborhood graph. Essentially,
the algorithmic process preserves local isometry between nearest neighbors
allowing distant neighbors to excurse expansively by “maximizing variance”
(Figure 5). The common number of nearest neighbors to each sample is
a data-dependent algorithmic parameter whose minimum value connects
the graph. The dimensionless EDM subgraph between each sample and
its nearest neighbors is completed from available data and included as
input; one such EDM subgraph completion is drawn superimposed upon the
neighborhood graph in Figure 116. 6.3 The consequent dimensionless EDM
graph comprising all the subgraphs is incomplete, in general, because the
neighbor number is relatively small; incomplete even though it is a superset
of the neighborhood graph. Remaining distances (those not graphed at all)
are squared then made variables within the algorithm; it is this variability
that admits unfurling.
To demonstrate, consider untying the trefoil knot drawn in Figure 117(a).
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 1N
ď 12 0 ď 23 ď 24
... ? ? ď 2N
ď 13 ď 23 0 ď 34
... ? ? ?
D =
? ď 24 ď 34 0
...
... ? ?
(1090)
.
...
...
...
...
...
... ?
? ? ?
...
... 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 0
where ďij denotes a given fixed distance-square. The unfurling algorithm
can be expressed as an optimization problem; constrained distance-square
6.3 Local reconstruction of point position from the EDM submatrix corresponding to a
complete dimensionless EDM subgraph is unique to within an isometry (5.6,5.12).