v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization 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).
6.4. A GEOMETRY OF COMPLETION 397(a)(b)Figure 97: (a) Trefoil knot in R 3 from Weinberger & Saul [278].(b) Topological transformation algorithm employing 4 nearest neighbors andN = 539 samples reduces affine dimension of knot to r=2. Choosing instead2 nearest neighbors would make this embedding more circular.
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- Page 361 and 362: 5.11. EDM INDEFINITENESS 361(confer
- Page 363 and 364: 5.11. EDM INDEFINITENESS 363we have
- Page 365 and 366: 5.11. EDM INDEFINITENESS 365For pre
- Page 367 and 368: 5.12. LIST RECONSTRUCTION 367where
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- Page 373 and 374: 5.13. RECONSTRUCTION EXAMPLES 373Th
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- Page 387 and 388: Chapter 6EDM coneFor N > 3, the con
- Page 389 and 390: 6.1. DEFINING EDM CONE 3896.1 Defin
- Page 391 and 392: 6.2. POLYHEDRAL BOUNDS 391This cone
- Page 393 and 394: 6.3.√EDM CONE IS NOT CONVEX 393N
- Page 395: 6.4. A GEOMETRY OF COMPLETION 3956.
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- Page 403 and 404: 6.5. EDM DEFINITION IN 11 T 403then
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- Page 419 and 420: 6.8. DUAL EDM CONE 419When the Fins
- Page 421 and 422: 6.8. DUAL EDM CONE 421Proof. First,
- Page 423 and 424: 6.8. DUAL EDM CONE 423EDM 2 = S 2 h
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- Page 427 and 428: 6.8. DUAL EDM CONE 4276.8.1.3.1 Exe
- Page 429 and 430: 6.8. DUAL EDM CONE 429has dual affi
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- Page 441 and 442: 441HS N h0EDM NK = S N h ∩ R N×N
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6.4. A GEOMETRY OF COMPLETION 397(a)(b)Figure 97: (a) Trefoil knot in R 3 from Weinberger & Saul [278].(b) Topological transformation algorithm employing 4 nearest neighbors andN = 539 samples reduces affine dimension of knot to r=2. Choosing instead2 nearest neighbors would make this embedding more circular.
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