Read e-book online Algorithms for minimization without derivatives PDF

By Richard P. Brent

ISBN-10: 0130223352

ISBN-13: 9780130223357

ISBN-10: 0486419983

ISBN-13: 9780486419985

Extraordinary textual content for graduate scholars and learn employees proposes advancements to current algorithms, extends their comparable mathematical theories, and provides info on new algorithms for approximating neighborhood and international minima. Many numerical examples, in addition to entire research of fee of convergence for many of the algorithms and mistake bounds that permit for the impression of rounding errors.

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G is a sigmoid or sine activation function. Other notations are defined in Table 1. 2 Basic-ELM For M arbitrary distinct samples (xi , ti ), where xi = [xi1 , xi2 , … , xin ]T ∈ Rm and ti ∈ R. ELM is proposed for SLFNs and output function of ELM for SLFNs is fl (x) = L ∑ ????i g(ai ⋅ xj + bi ) = H ⋅ ???? (1) i=1 where ???? is the output weight matrix connecting hidden nodes to output nodes, g represents an activation function. , y ∈ Rm×N Feature data dimension (2) 34 Y. J. Wu For L hidden nodes, H is referred to as ELM feature mapping or Huang’s transform: ⎡ g(x1 ) ⎤ ⎡ g1 (x1 ) ⋯ gL (x1 ) ⎤ H=⎢ ⋮ ⎥=⎢ ⋮ ⋯ ⋮ ⎥ ⎥ ⎢ ⎥ ⎢ ⎣g(xM )⎦ ⎣g1 (xM ) ⋯ gL (xM )⎦ (3) and t is the training data target matrix: ⎡ t1T ⎤ ⎡ t11 ⋯ t1m ⎤ t = ⎢⋮⎥ = ⎢ ⋮ ⋯ ⋮ ⎥ ⎥ ⎢T⎥ ⎢ ⎣tM ⎦ ⎣tM1 ⋯ tMm ⎦ (4) Huang et al.

Both ELM and PC-ELM run much faster than the SVM method as the feature dimensionality increases, as illustrated in Fig. 5b. 28 (a) The average accuracy of 50 trails classification experiments. 86 84 82 80 performance (%) Fig. 5 Performance results of PHOW. The x-axis indicates the number of engaged features applied for the experiments. The y-axis denotes the average test set classification accuracy results in (a). In b, the y-axis indicates the training time. Note that this figure is represented in semi-logarithmic coordinates D.

4(b), respectively; moreover, the corresponding exact values of RMSE after 20 times iteration are written down in Table 1. 24 0 5 10 15 20 Iterations Fig. 4 The values of RMSE with respect to different number of processors. a Training.

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