v2009.01.01 - Convex Optimization
v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization
452 CHAPTER 6. CONE OF DISTANCE MATRICES That is easily proven false by counter-example via (1088), for then ( ◦√ D 1 + ◦√ D 2 ) ◦( ◦√ D 1 + ◦√ D 2 ) would need to be a member of EDM N . Notwithstanding, √ EDM N ⊆ EDM N (1089) by (999) (Figure 114), and we learn how to transform a nonconvex proximity problem in the natural coordinates √ d ij to a convex optimization in7.2.1. 6.4 a geometry of completion Intriguing is the question of whether the list in X ∈ R n×N (68) may be reconstructed given an incomplete noiseless EDM, and under what circumstances reconstruction is unique. [1] [3] [4] [5] [7] [16] [179] [188] [201] [202] [203] If one or more entries of a particular EDM are fixed, then geometric interpretation of the feasible set of completions is the intersection of the EDM cone EDM N in isomorphic subspace R N(N−1)/2 with as many hyperplanes as there are fixed symmetric entries. (Depicted in Figure 115(a) is an intersection of the EDM cone EDM 3 with a single hyperplane representing the set of all EDMs having one fixed symmetric entry.) Assuming a nonempty intersection, then the number of completions is generally infinite, and those corresponding to particular affine dimension r
6.4. A GEOMETRY OF COMPLETION 453 (a) 2 nearest neighbors (b) 3 nearest neighbors Figure 116: Neighborhood graph (dashed) with dimensionless EDM subgraph completion (solid) superimposed (but not covering dashed). Local view of a few dense samples from relative interior of some arbitrary Euclidean manifold whose affine dimension appears two-dimensional in this neighborhood. All line segments measure absolute distance. Dashed line segments help visually locate nearest neighbors; suggesting, best number of nearest neighbors can be greater than value of embedding dimension after topological transformation. (confer [183,2]) Solid line segments represent completion of EDM subgraph from available distance data for an arbitrarily chosen sample and its nearest neighbors. Each distance from EDM subgraph becomes distance-square in corresponding EDM submatrix.
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- Page 419 and 420: 5.11. EDM INDEFINITENESS 419 we hav
- Page 421 and 422: 5.11. EDM INDEFINITENESS 421 So bec
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- Page 445 and 446: Chapter 6 Cone of distance matrices
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6.4. A GEOMETRY OF COMPLETION 453<br />
(a)<br />
2 nearest neighbors<br />
(b)<br />
3 nearest neighbors<br />
Figure 116: Neighborhood graph (dashed) with dimensionless EDM subgraph<br />
completion (solid) superimposed (but not covering dashed). Local view<br />
of a few dense samples from relative interior of some arbitrary<br />
Euclidean manifold whose affine dimension appears two-dimensional in this<br />
neighborhood. All line segments measure absolute distance. Dashed line<br />
segments help visually locate nearest neighbors; suggesting, best number of<br />
nearest neighbors can be greater than value of embedding dimension after<br />
topological transformation. (confer [183,2]) Solid line segments represent<br />
completion of EDM subgraph from available distance data for an arbitrarily<br />
chosen sample and its nearest neighbors. Each distance from EDM subgraph<br />
becomes distance-square in corresponding EDM submatrix.
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