Download Discrete Mathematics of Neural Networks: Selected Topics by Martin Anthony PDF

By Martin Anthony

This concise, readable booklet offers a sampling of the very huge, energetic, and increasing box of synthetic neural community conception. It considers decide on parts of discrete arithmetic linking combinatorics and the idea of the best forms of man made neural networks. Neural networks have emerged as a key expertise in lots of fields of program, and an realizing of the theories bearing on what such platforms can and can't do is vital.

The writer discusses fascinating connections among precise varieties of Boolean services and the easiest different types of neural networks. a few classical effects are awarded with available proofs, including a few newer views, corresponding to these received through contemplating selection lists. moreover, probabilistic versions of neural community studying are mentioned. Graph conception, a few partly ordered set concept, computational complexity, and discrete chance are one of the mathematical issues concerned. tips to additional analyzing and an intensive bibliography make this publication an exceptional place to begin for study in discrete arithmetic and neural networks.

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Extra resources for Discrete Mathematics of Neural Networks: Selected Topics (Monographs on Discrete Mathematics and Applications)

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5 For all n, the number, \Tn\, of threshold functions on {0, l}n satisfies Proof. The crucial part of this proof is showing that \Tn\ > (2 n ~ 1 + l)|T n _i| for all n > I. = /. Suppose that / G Tn-\. Suppose that the weight vector w G Rn~1 and threshold 0 are such that / 4- [w, 9] and also such that the numbers WTX, as a: ranges through (0,1}™"1, are distinct. ) Suppose that w' = (w, wn) G R n , where wn G R, and let g be the threshold function in Tn represented by weight-vector w' and threshold 0.

The upper bound has been observed by a number of researchers; see [29]. 5 is due to Muroga [75, 74]. The asymptotic behavior of Iog2 \Tn\ was determined by Zuev [106]. 3 appears in [20], and the proof presented here is from [26]. 7, see [13, 5]. 8, is due to Saks [90]. 1 Introduction A weight-vector and threshold are said to be integral if the threshold and each entry of the weight-vector are integers. Any Boolean threshold function can be represented by an integral weight-vector and threshold.

Suppose then that / was a threshold function, represented by weight vector w € R10 and threshold 9. Now, we note that if a = 0111110101, 6=1011111010, c = 1111110000, d = 0011111111, then This means that But we should then have However, this cannot be, since a + b = c + d. We therefore have a contradiction, and it follows that / is not a threshold function. This example demonstrates some interesting features of threshold functions. First, the conjunction of threshold functions is not necessarily a threshold function.

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