给初学者看的神经网络的数学基础 A Beginner's Guide to the Mathematics of Neural ...
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A Beginner's Guide to the
Mathematics of Neural Networks
A.C.C. Co olen
Department of Mathematics, King's College London
Abstract
In this pap er I try to describe both the role of mathematics in shap-
ing our understanding of how neural networks operate, and the curious
new mathematical concepts generated by our attempts to capture neu-
ral networks in equations. My target reader b eing the non-exp ert, I will
present a biased selection of relatively simple examples of neural network
tasks, models and calculations, rather than try to give a full encyclop edic
review-like account of the many mathematical developments in this eld.
Contents
1 Intro duction: Neural Information Pro cessing 2
2 From Biology to Mathematical Mo dels 6
2.1 From Biological Neurons to Mo del Neurons . . . . . . . . . . . 6
2.2 Universality of Mo del Neurons . . . . . . . . . . . . . . . . . . 9
2.3 Directions and Strategies . . . . . . . . . . . . . . . . . . . . . 12
3 Neural Networks as Asso ciative Memories 14
3.1 Recipes for Storing Patterns and Pattern Sequences . . . . . . 15
3.2 Symmetric Networks: the Energy Picture . . . . . . . . . . . . 19
3.3 Solving Mo dels of Noisy Attractor Networks . . . . . . . . . . . 20
4 Creating Maps of the Outside World 26
4.1 Map Formation Through Competitive Learning . . . . . . . . . 26
4.2 Solving Mo dels of Map Formation . . . . . . . . . . . . . . . . 29
5 Learning a Rule From an Exp ert 35
5.1 Perceptrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Multi-layer Networks . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3 Calculating what is Achievable . . . . . . . . . . . . . . . . . . 43
5.4 Solving the Dynamics of Learning for Perceptrons . . . . . . . 47
6 Puzzling Mathematics 52
6.1 Complexity due to Frustration, Disorder and Plasticity. . . . . 52
6.2 The World of Replica Theory . . . . . . . . . . . . . . . . . . . 55
7 Further Reading 59
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