支持向量机与核方法学习导论

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"《支持向量机导论》是由Nello Cristianini和John Shawe-Taylor合著的一本深入介绍支持向量机(SVM)及其相关核基学习方法的著作。该书由剑桥大学出版社于2000年出版，共190页，旨在帮助读者理解这种基于统计学习理论的新一代学习系统的理论基础及其实际应用。书中包含了从学习方法论到线性学习机器，再到核函数诱导的特征空间，以及泛化理论、优化理论、SVM的具体实现技术和各种SVM的应用实例。此外，还有关于SMO算法的伪代码、背景数学知识以及参考文献和索引等补充材料。" 支持向量机（Support Vector Machines，简称SVM）是一种监督学习模型，主要应用于分类和回归分析。SVM的核心思想是找到一个最优超平面，将不同类别的数据点分隔开来，同时最大化这个超平面与最近的数据点（支持向量）之间的间隔。这种模型在处理小样本和高维数据时表现出色，尤其在面对非线性可分问题时，通过使用核函数，可以将数据映射到高维空间，实现线性分类。在《支持向量机导论》一书中，作者首先介绍了学习方法论，阐述了学习的基本原理。接着，他们讨论了线性学习机器，包括线性分类器的基础知识。然后，引入了核函数的概念，这是SVM能够处理非线性问题的关键，通过核函数，原始数据被转换为一个新的特征空间，在这个空间内可能更容易找到分类超平面。第四章的泛化理论探讨了模型的预测能力，解释了为什么SVM在防止过拟合方面表现优秀。第五章的优化理论则深入研究了求解SVM的最优化问题，包括拉格朗日乘子法和凸优化技术。第六章专门讲解支持向量机，详细介绍了SVM的构造、训练过程以及如何选择合适的核函数。第七章涵盖了实施技巧，如如何解决大规模数据集的问题，以及如何调整SVM的参数。第八章列举了SVM在各个领域的应用，如文本分类、生物信息学、计算机视觉等，展示了SVM在实际问题中的强大适应性和有效性。附录部分提供了SMO（Sequential Minimal Optimization）算法的伪代码，这是求解SVM最优化问题的一种高效算法。同时，还包含了必要的背景数学知识，以便读者更好地理解SVM背后的理论基础。《支持向量机导论》是一本全面且深入的SVM学习资料，无论对于初学者还是经验丰富的从业者，都能从中获得宝贵的理论知识和实践指导。

If no restriction is placed over the set of all possible hypotheses (that is all possible functions from the input

space to the output domain), then learning is impossible since no amount of training data will tell us how to

classify unseen examples. Problems also arise if we allow ourselves the freedom of choosing the set of

hypotheses after seeing the data, since we can simply assume all of the prior probability on the correct

hypotheses. In this sense it is true that all learning systems have to make some prior assumption of a Bayesian

type often called the learning bias. We will place the approach we adopt in this context in Chapter 4.

2 Chapter 1: The Learning Methodology

1.4 Attractions and Drawbacks of Learning

It is not surprising that the promise of the learning methodology should be so tantalising. Firstly, the range of

applications that can potentially be solved by such an approach is very large. Secondly, it appears that we can

also avoid much of the laborious design and programming inherent in the traditional solution methodology, at

the expense of collecting some labelled data and running an off-the-shelf algorithm for learning the

input/output mapping. Finally, there is the attraction of discovering insights into the way that humans

function, an attraction that so inspired early work in neural networks, occasionally to the detriment of

scientific objectivity.

There are, however, many difficulties inherent in the learning methodology, difficulties that deserve careful

study and analysis. One example is the choice of the class of functions from which the input/output mapping

must be sought. The class must be chosen to be sufficiently rich so that the required mapping or an

approximation to it can be found, but if the class is too large the complexity of learning from examples can

become prohibitive, particularly when taking into account the number of examples required to make

statistically reliable inferences in a large function class. Hence, learning in three-node neural networks is

known to be NP-complete, while the problem of minimising the number of training errors of a thresholded

linear function is also NP-hard. In view of these difficulties it is immediately apparent that there are severe

restrictions on the applicability of the approach, and any suggestions of a panacea are highly misleading.

In practice these problems manifest themselves in specific learning difficulties. The first is that the learning

algorithm may prove inefficient as for example in the case of local minima. The second is that the size of the

output hypothesis can frequently become very large and impractical. The third problem is that if there are only

a limited number of training examples too rich a hypothesis class will lead to overfitting and hence poor

generalisation. The fourth problem is that frequently the learning algorthm is controlled by a large number of

parameters that are often chosen by tuning heuristics, making the system difficult and unreliable to use.

Despite the drawbacks, there have been notable successes in the application of the learning methodology to

problems of practical interest. Unfortunately, however, there is frequently a lack of understanding about the

conditions that will render an application successful, in both algorithmic and statistical inference terms. We

will see in the next section that Support Vector Machines address all of these problems.

Chapter 1: The Learning Methodology 1

1.5 Support Vector Machines for Learning

Support Vector Machines (SVM) are learning systems that use a hypothesis space of linear functions in a high

dimensional feature space, trained with a learning algorithm from optimisation theory that implements a

learning bias derived from statistical learning theory. This learning strategy introduced by Vapnik and

co-workers is a principled and very powerful method that in the few years since its introduction has already

outperformed most other systems in a wide variety of applications.

This book gives an introduction to Support Vector Machines by describing the hypothesis space and its

representation in Chapters 2 and 3, the learning bias in Chapter 4, and the learning algorithm in Chapters 5

and 7. Chapter 6 is the key chapter in which all these components are brought together, while Chapter 8 gives

an overview of some applications that have been made to real-world problems. The book is written in a

modular fashion so that readers familiar with the material covered in a particular chapter can safely bypass

that section. In particular, if the reader wishes to get a direct introduction to what an SVM is and how to

implement one, they should move straight to Chapters 6 and 7.

Chapter 2 introduces linear learning machines one of the main building blocks of the system. Chapter 3 deals

with kernel functions which are used to define the implicit feature space in which the linear learning machines

operate. The use of kernel functions is the key to the efficient use of high dimensional feature spaces. The

danger of overfitting inherent in high dimensions requires a sophisticated learning bias provided by the

statistical learning theory covered in Chapter 4. Optimisation theory covered in Chapter 5 gives a precise

characterisation of the properties of the solution which guide the implementation of efficient learning

algorithms described in Chapter 7, and ensure that the output hypothesis has a compact representation. The

particular choice of a convex learning bias also results in the absence of local minima so that solutions can

always be found efficiently even for training sets with hundreds of thousands of examples, while the compact

representation of the hypothesis means that evaluation on new inputs is very fast. Hence, the four problems of

efficiency of training, efficiency of testing, overfitting and algorithm parameter tuning are all avoided in the

SVM design.

Chapter 1: The Learning Methodology 1

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支持向量机与核方法学习导论

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