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Table 5.4 – Inner-Outer preconditioned BiCG-STAB performance. Matrix-vector products required for BiCG-STAB on in-2004, including preconditioning and residual computations, to converge on the system (I − αP ̄ ) with preconditioner ∑m (βP)k . A dash indicates the method made k=0 √ progress but did not converge to a tolerance of ( 1 − α)10−7 in the maximum number of iterations required for the power method (100 for α = 0.85, ≈1500 for α = 0.99, ≈15000 for α = 0.999), and an × indicates the method diverged or broke-down. When m = 0, there is no preconditioning and the results are independent of β. α 0.85 0.99 0.999 m 0.25 0 102 2 128 4 186 7 — 25— 0.50 0.75 0.85 102 102 102 88 76 76 120 84 78 207 108 72 — — 81 0.25 0.50 0.75 0.85 0.25 0.50 0.75 0.85 5.6 ⋅ numerical results 125 βββ ×××× 1140 672 508 500 1584 786 438 414 2565 1053 621 441 — — 1809 1026 ×××× × 6276 3972 2772 × 5178 2358 2112 × 9567 2709 1449 — 20385 7911 2754 Let P and Q be the uniform random-walk transition matrices for A and B, respectively. Also, let the weights in V be normalized so that eT Ve = 1 and Vi j ≥ 0. IsoRank uses the PageRank vector x = α(P ⊗ Q)x + (1 − α)v, where the teleportation vector v = vec(V) encodes the weights and α indi- cates how much emphasis to place on matches using the weights information. Thus the IsoRank algorithm is a case when v is not uniform, and α has a more concrete meaning. For a protein matching problem, it is observed experimen- tally in Singh et al. [2007] that values of α between 0.7 and 0.95 yield good results. We look at a case when A is the 2-core of the undirected graph of subject headings from the Library of Congress [Various, 2008] (abbreviated LCSH-2) and B is the 3-core of the undirected Wikipedia category structure [Various, 2007] (abbreviated WC-3). We previously used these datasets in analyzing the actual matches in a slightly different setting [Bayati et al., 2009]. The size of these datasets is reported in table 5.5. For this application, the weights come from a text-matching procedure on the labels of the two graphs. dataset LCSH-2 WC-3 Product Graph size 59,849 70,509 4,219,893,141 non-zeros 227,464 403,960 91,886,357,440 Table 5.5 – IsoRank datasets.PDF Image | Instagram Cheat Sheet
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