A Beginners Guide to the Mathematics of Neural Networks
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A Beginners Guide to the Mathematics of Neural Networks
A Beginner’s Guide to the Mathematics of Neural NetworksA.c.c. CoolenDepartment of Mathematics. King's College LondonAbstractIn this ỊNiper I try to d A Beginners Guide to the Mathematics of Neural Networks describe both the role of mathematics in shaping our understanding of how neural networks operate, ami the curious new mathematical concepts generated by our attempts to capture neural networks in equations. My target reader being the nou-expert. I will present a biased selection of relatively simpl A Beginners Guide to the Mathematics of Neural Networks e examples of neural network tasks, models ami calculations, rather than try to give a full encyclopedic review-like account of the many mathematicalA Beginners Guide to the Mathematics of Neural Networks
developments in this field.Contents1Introduction: Neural Information Processing22From Biology to Mathematical Models62.1From Biological Neurons to ModA Beginner’s Guide to the Mathematics of Neural NetworksA.c.c. CoolenDepartment of Mathematics. King's College LondonAbstractIn this ỊNiper I try to d A Beginners Guide to the Mathematics of Neural Networks .... 123Neural Networks as Associative Memories143.1Recipes for Storing Patterns and Pattern Sequences.......153.2Symmetric Networks: the Energy Picture................... 193.3Solving Models of Noisy Attractor Networks............... 204Creating Maps of the Outside World264.1Map Formation Through C A Beginners Guide to the Mathematics of Neural Networks ompetitive learning................ 264.2Solving Models of Map Formation ......................... 295Learning a Rule From an Expert355.1Perceptrons..A Beginners Guide to the Mathematics of Neural Networks
............................................. 355.2Multi-layer Networks..................................... 395.3Calculating what is Achievable......A Beginner’s Guide to the Mathematics of Neural NetworksA.c.c. CoolenDepartment of Mathematics. King's College LondonAbstractIn this ỊNiper I try to d A Beginners Guide to the Mathematics of Neural Networks r and Plasticity...526.2The World of Replica Theory.............................. 557Further Reading5911 Introduction: Neural Information ProcessingOur brains perform sophisticated information processing tasks, using hardware and operation rules which are quite different from the ones on which conve A Beginners Guide to the Mathematics of Neural Networks ntional computers are based. The processors in the brain, the neurons (see figure 1). are rather noisy elements1 which operate in parallel. They are oA Beginners Guide to the Mathematics of Neural Networks
rganised in dense networks, the structure of which can vary from very regular to almost amorphous (see figtire 2). and they communicate signals througA Beginner’s Guide to the Mathematics of Neural NetworksA.c.c. CoolenDepartment of Mathematics. King's College LondonAbstractIn this ỊNiper I try to d A Beginners Guide to the Mathematics of Neural Networks the strengths of the connections, a network as a whole can modify ami optimise its ‘program’, ■learn' from experience and adapt to changing circumstances.Figure 1: Left: a Purkinje neuron in the human cerebellum Hight: a pyramidal neuron of the rabbit cortex. The black blobs are the neurons, the tr A Beginners Guide to the Mathematics of Neural Networks ees of wires fanning out constitute the input channels (or dendrites) through which signals are received which are sent off by other firing neurons. TA Beginners Guide to the Mathematics of Neural Networks
he lines at the bottom, bifurcating only modestly, are the output channels (or axons).From an engineering point of view neurons are in fact rather pooA Beginner’s Guide to the Mathematics of Neural NetworksA.c.c. CoolenDepartment of Mathematics. King's College LondonAbstractIn this ỊNiper I try to d A Beginners Guide to the Mathematics of Neural Networks are involved in any task, and by having them operate in parallel, with many connections. This is in sharp contrast to conventional computers, where operations are as a rule performed sequentially, so that, failure of any part of the chain of operations is usually fatal. Furthermore, conventional co A Beginners Guide to the Mathematics of Neural Networks mputers execute a detailed specification of orders, requiring the programmer to know exactly which data can be expected and how to respond. SubsequentA Beginners Guide to the Mathematics of Neural Networks
changes in the actual situation, not foreseen by the programmer, lead to trouble. Neural networks, on the other hand, can adapt to changing circumstaA Beginner’s Guide to the Mathematics of Neural NetworksA.c.c. CoolenDepartment of Mathematics. King's College LondonAbstractIn this ỊNiper I try to d A Beginners Guide to the Mathematics of Neural Networks in our workstation.1 By this mean that their output signals are to Mime degree Subject to random variat ion: they exhibit so-called spontaneous activity which appears not to bl- related to the information processing task they arc involved in.2Figure 2: Left: a section of the human cerebellum. Right A Beginners Guide to the Mathematics of Neural Networks : a section of the human cortex. Note that the staining method used to produce such pictures colours only a reasonably modest fraction of the neuronsA Beginners Guide to the Mathematics of Neural Networks
present, so in reality these networks arc far more dense.Roughly speaking, conventional computers can be seen as the appropriate tools for performing A Beginner’s Guide to the Mathematics of Neural NetworksA.c.c. CoolenDepartment of Mathematics. King's College LondonAbstractIn this ỊNiper I try to d A Beginners Guide to the Mathematics of Neural Networks every situation, are known beforehand. Typical tasks fitting these criteria are e.g brute-force chess playing, word processing, keeping accounts and rule-based (civil servant) decision making. Neural information processing systems, oil the other hand, are superior to conventional computers ill deal A Beginners Guide to the Mathematics of Neural Networks ing with real-world tasks, such as e.g. communication (vision, speech recognition), Ifio-. «-- i. nr I III II dil ation (robotics) and experience baseA Beginners Guide to the Mathematics of Neural Networks
d decision making (clas sification, prediction, system control), where data are often messy, uncertain or oven inconsistent, where the number of possiA Beginner’s Guide to the Mathematics of Neural NetworksA.c.c. CoolenDepartment of Mathematics. King's College LondonAbstractIn this ỊNiper I try to dA Beginner’s Guide to the Mathematics of Neural NetworksA.c.c. CoolenDepartment of Mathematics. King's College LondonAbstractIn this ỊNiper I try to dGọi ngay
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