Discrete Mathematics of Neural Networks: Selected Topics
This concise, readable book provides a sampling of the very large, active, and expanding field of artificial neural network theory. It considers select areas of discrete mathematics linking combinatorics and the theory of the simplest types of artificial neural networks. Neural networks have emerged as a key technology in many fields of application, and an understanding of the theories concerning what such systems can and cannot do is essential. Some classical results are presented with accessible proofs, together with some more recent perspectives, such as those obtained by considering decision lists. In addition, probabilistic models of neural network learning are discussed. Graph theory, some partially ordered set theory, computational complexity, and discrete probability are among the mathematical topics involved. Pointers to further reading and an extensive bibliography make this book a good starting point for research in discrete mathematics and neural networks.
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2-monotonic antichain artificial neural networks Boltzmann machine Boolean function Boolean hypercube Boolean threshold functions Combinatorial computation unit Computational Learning Theory consistent learning algorithm convex defined denote Discrete Mathematics DNF formula DNF representation false points feed-forward finite following result function f function of degree functions computable given graph hidden units hypercube hyperplane hypothesis space identically-1 function increasing function inequality inputs integer linear programming linear threshold function linear threshold unit log2 lower bound MFP(t MTP(t MTPs MTPs of f multisets nested function neural network NP-complete NP-hard number of threshold obtain output function output unit PAC learning algorithm PARITYn perceptron polynomial threshold functions positive example prime implicants problem proof of Theorem remarks and bibliographical shattered simply specification number specifying set subsets Suppose target function Theory threshold order total number training sample true points two-layer upper bound variables VC-dimension vector