Get Artificial Neural Networks in Pattern Recognition: 7th IAPR PDF

By Friedhelm Schwenker, Hazem M. Abbas, Neamat El Gayar, Edmondo Trentin

ISBN-10: 3319461818

ISBN-13: 9783319461816

ISBN-10: 3319461826

ISBN-13: 9783319461823

This ebook constitutes the refereed court cases of the seventh IAPR TC3 overseas Workshop on man made Neural Networks in development acceptance, ANNPR 2016, held in Ulm, Germany, in September 2016.

The 25 revised complete papers offered including 2 invited papers have been conscientiously reviewed and chosen from 32 submissions for inclusion during this quantity. The workshop will act as an enormous discussion board for overseas researchers and practitioners operating in all parts of neural community- and computer learning-based development acceptance to provide and speak about the most recent examine, effects, and ideas in those areas.

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Extra info for Artificial Neural Networks in Pattern Recognition: 7th IAPR TC3 Workshop, ANNPR 2016, Ulm, Germany, September 28–30, 2016, Proceedings

Sample text

This work was supported by JSPS KAKENHI Grant Number 25420438. References 1. : Statistical Learning Theory. Wiley, New York (1998) 2. : Support Vector Machines for Pattern Classification, 2nd edn. Springer, London (2010) 40 S. Abe 3. : A robust minimax approach to classification. J. Mach. Learn. Res. 3, 555–582 (2002) 4. : Learning large margin classifiers locally and globally. In: Proceedings of the Twenty-First International Conference on Machine Learning (ICML 2004), pp. 1–8 (2006) 5. : Structured large margin machines: sensitive to data distributions.

By the triangle inequality we get ˆ −K K F ≤ ϕ ϕ−K F ˜ + ϕ ϕ−K F (9) ˜ is the Nystr¨ where K om approximated kernel matrix of the linear kernel matrix on the random fourier features and ϕ ϕ is given as ϕ(xi ) ϕ(xj ) = α D D cos(ωl (xi − xj )) (10) l=1 where D is the number of random fourier features ωl ∼ N (0, Id ), α > 0 and xi , xj are training points with RFF feature values stored in ϕ. In the following we derive and combine the bounds for both approximation schemes. The Frobenius error of the approximated kernel using the random fourier features is given as N ϕ ϕ−K F N |ϕ(xi ) ϕ(xj ) − K(xi , xj )|2 = (11) i=1 j=1 with K as the kernel matrix of the training points {x1 , .

Hence we can use bounds already provided in [24]. -M. Schleif et al. 1 with β(m, n) = 1 − 2 max m,N −m and Kk the best k approximation of K and Kmax the maximal diagonal entry of K and dK max the maximum Euclidean distance defined over K. Which maybe summarized in accordance to [22] as D 14 ϕ ϕ − Kk F + [ m ] N K 2 . Combining both bounds we get ˆ −K K F ≤ ϕ ϕ−K ≤ ϕ ϕ − Kk N· α 2 D F ˜ + ϕ ϕ−K F F + log 2 + 2 log N δ +[ D 1 ]4 K m 2 (18) ˜ ) - that is, up to log facWe see that both approximation terms increase as O(N tors the kernel approximation error increases linearly with the number of training points N .

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Artificial Neural Networks in Pattern Recognition: 7th IAPR TC3 Workshop, ANNPR 2016, Ulm, Germany, September 28–30, 2016, Proceedings by Friedhelm Schwenker, Hazem M. Abbas, Neamat El Gayar, Edmondo Trentin

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