The last decade has seen an explosion of readily available digital text that has rendered attempts to analyse and classify by hand infeasible. As a result automatic processing of natural language text documents has become a main research interest of Artificial Intelligence (AI) and computer science in general. It is probably fair to say that after multivariate data, natural language text is the most important data format for applications. Its particular characteristics therefore deserve specific attention.

We will see how well-known techniques from Information Retrieval (IR), such as the rich class of vector space models, can be naturally reinterpreted as kernel methods. This new perspective enriches our understanding of the approach, as well as leading naturally to further extensions and improvements. The approach that this perspective suggests is based on detecting and exploiting statistical patterns of words in the documents. An important property of the vector space representation is that the primal–dual dialectic we have developed through this book has an interesting counterpart in the interplay between term-based and document-based representations.

The goal of this chapter is to introduce the Vector Space family of kernel methods highlighting their construction and the primal–dual dichotomy that they illustrate. Other kernel constructions can be applied to text, for example using probabilistic generative models and string matching, but since these kernels are not specific to natural language text, they will be discussed separately in Chapters 11 and 12.

The previous chapter saw the development of some basic tools for working in a kernel-defined feature space resulting in some useful algorithms and techniques. The current chapter will extend the methods in order to understand the spread of the data in the feature space. This will be followed by examining the problem of identifying correlations between input vectors and target values. Finally, we discuss the task of identifying covariances between two different representations of the same object.

All of these important problems in kernel-based pattern analysis can be reduced to performing an eigen- or generalised eigen-analysis, that is the problem of finding solutions of the equation Aw = λBw given symmetric matrices A and B. These problems range from finding a set of k directions in the embedding space containing the maximum amount of variance in the data (principal components analysis (PCA)), through finding correlations between input and output representations (partial least squares (PLS)), to finding correlations between two different representations of the same data (canonical correlation analysis (CCA)). Also the Fisher discriminant analysis from Chapter 5 can be cast as a generalised eigenvalue problem.

The importance of this class of algorithms is that the generalised eigenvectors problem provides an efficient way of optimising an important family of cost functions; it can be studied with simple linear algebra and can be solved or approximated efficiently using a number of well-known techniques from computational algebra.

In this chapter we conclude our presentation of kernel-based pattern analysis algorithms by discussing three further common tasks in data analysis: ranking, clustering and data visualisation.

Ranking is the problem of learning a ranking function from a training set of ranked data. The number of ranks need not be specified though typically the training data comes with a relative ordering specified by assignment to one of an ordered sequence of labels.

Clustering is perhaps the most important and widely used method of unsupervised learning: it is the problem of identifying groupings of similar points that are relatively ‘isolated’ from each other, or in other words to partition the data into dissimilar groups of similar items. The number of such clusters may not be specified a priori. As exact solutions are often computationally hard to find, effective approximations via relaxation procedures need to be sought.

Data visualisation is often overlooked in pattern analysis and machine learning textbooks, despite being very popular in the data mining literature. It is a crucial step in the process of data analysis, enabling an understanding of the relations that exist within the data by displaying them in such a way that the discovered patterns are emphasised. These methods will allow us to visualise the data in the kernel-defined feature space, something very valuable for the kernel selection process. Technically it reduces to finding low-dimensional embeddings of the data that approximately retain the relevant information.

There are two key properties that are required of a kernel function for an application. Firstly, it should capture the measure of similarity appropriate to the particular task and domain, and secondly, its evaluation should require significantly less computation than would be needed in an explicit evaluation of the corresponding feature mapping ϕ. Both of these issues will be addressed in the next four chapters but the current chapter begins the consideration of the efficiency question.

A number of computational methods can be deployed in order to shortcut the computation: some involve using closed-form analytic expressions, others exploit recursive relations, and others are based on sampling. This chapter aims to show several different methods in action, with the aim of illustrating how to design new kernels for specific applications. It will also pave the way for the final three chapters that carry these techniques into the design of advanced kernels.

We will also return to an important theme already broached in Chapter 3, namely that kernel functions are not restricted to vectorial inputs: kernels can be designed for objects and structures as diverse as strings, graphs, text documents, sets and graph-nodes. Given the different evaluation methods and the diversity of the types of data on which kernels can be defined, together with the methods for composing and manipulating kernels outlined in Chapter 3, it should be clear how versatile this approach to data modelling can be, allowing as it does for refined customisations of the embedding map ϕ to the problem at hand.

Pattern analysis deals with the automatic detection of patterns in data, and plays a central role in many modern artificial intelligence and computer science problems. By patterns we understand any relations, regularities or structure inherent in some source of data. By detecting significant patterns in the available data, a system can expect to make predictions about new data coming from the same source. In this sense the system has acquired generalisation power by ‘learning’ something about the source generating the data. There are many important problems that can only be solved using this approach, problems ranging from bioinformatics to text categorization, from image analysis to web retrieval. In recent years, pattern analysis has become a standard software engineering approach, and is present in many commercial products.

Early approaches were efficient in finding linear relations, while nonlinear patterns were dealt with in a less principled way. The methods described in this book combine the theoretically well-founded approach previously limited to linear systems, with the flexibility and applicability typical of nonlinear methods, hence forming a remarkably powerful and robust class of pattern analysis techniques.

There has been a distinction drawn between statistical and syntactical pattern recognition, the former dealing essentially with vectors under statistical assumptions about their distribution, and the latter dealing with structured objects such as sequences or formal languages, and relying much less on statistical analysis.

In Chapter 1 we gave a general overview to pattern analysis. We identified three properties that we expect of a pattern analysis algorithm: computational efficiency, robustness and statistical stability. Motivated by the observation that recoding the data can increase the ease with which patterns can be identified, we will now outline the kernel methods approach to be adopted in this book. This approach to pattern analysis first embeds the data in a suitable feature space, and then uses algorithms based on linear algebra, geometry and statistics to discover patterns in the embedded data.

The current chapter will elucidate the different components of the approach by working through a simple example task in detail. The aim is to demonstrate all of the key components and hence provide a framework for the material covered in later chapters.

Any kernel methods solution comprises two parts: a module that performs the mapping into the embedding or feature space and a learning algorithm designed to discover linear patterns in that space. There are two main reasons why this approach should work. First of all, detecting linear relations has been the focus of much research in statistics and machine learning for decades, and the resulting algorithms are both well understood and efficient. Secondly, we will see that there is a computational shortcut which makes it possible to represent linear patterns efficiently in high-dimensional spaces to ensure adequate representational power. The shortcut is what we call a kernel function.

This chapter presents a number of algorithms for particular pattern analysis tasks such as novelty-detection, classification and regression. We consider criteria for choosing particular pattern functions, in many cases derived from stability analysis of the corresponding tasks they aim to solve. The optimisation of the derived criteria can be cast in the framework of convex optimization, either as linear or convex quadratic programs. This ensures that as with the algorithms of the last chapter the methods developed here do not suffer from the problem of local minima. They include such celebrated methods as support vector machines for both classification and regression.

We start, however, by describing how to find the smallest hypersphere containing the training data in the embedding space, together with the use and analysis of this algorithm for detecting anomalous or novel data. The techniques introduced for this problem are easily adapted to the task of finding the maximal margin hyperplane or support vector solution that separates two sets of points again possibly allowing some fraction of points to be exceptions. This in turn leads to algorithms for the case of regression.

An important feature of many of these systems is that, while enforcing the learning biases suggested by the stability analysis, they also produce ‘sparse’ dual representations of the hypothesis, resulting in efficient algorithms for both training and test point evaluation. This is a result of the Karush–Kuhn–Tucker conditions, which play a crucial role in the practical implementation and analysis of these algorithms.

It is often the case that we know something about the process generating the data. For example, DNA sequences have been generated through evolution in a series of modifications from ancestor sequences, text can be viewed as being generated by a source of words perhaps reflecting the topic of the document, a time series may have been generated by a dynamical system of a certain type, 2-dimensional images by projections of a 3-dimensional scene, and so on.

For all of these data sources we have some, albeit imperfect, knowledge about the source generating the data, and hence of the type of invariances, features and similarities (in a word, patterns) that we can expect it to contain. Even simple and approximate models of the data can be used to create kernels that take account of the insight thus afforded.

Models of data can be either deterministic or probabilistic and there are also several different ways of turning them into kernels. In fact some of the kernels we have already encountered can be regarded as derived from generative data models.

However in this chapter we put the emphasis on generative models of the data that are frequently pre-existing. We aim to show how these models can be exploited to provide features for an embedding function for which the corresponding kernel can be efficiently evaluated.

Although the emphasis will be mainly on two classes of kernels induced by probabilistic models, P-kernels and Fisher kernels, other methods exist to incorporate generative information into kernel design.

As we have seen in Chapter 2, the use of kernel functions provides a powerful and principled way of detecting nonlinear relations using well-understood linear algorithms in an appropriate feature space. The approach decouples the design of the algorithm from the specification of the feature space. This inherent modularity not only increases the flexibility of the approach, it also makes both the learning algorithms and the kernel design more amenable to formal analysis. Regardless of which pattern analysis algorithm is being used, the theoretical properties of a given kernel remain the same. It is the purpose of this chapter to introduce the properties that characterise kernel functions.

We present the fundamental properties of kernels, thus formalising the intuitive concepts introduced in Chapter 2. We provide a characterization of kernel functions, derive their properties, and discuss methods for designing them. We will also discuss the role of prior knowledge in kernel-based learning machines, showing that a universal machine is not possible, and that kernels must be chosen for the problem at hand with a view to capturing our prior belief of the relatedness of different examples. We also give a framework for quantifying the match between a kernel and a learning task.

Given a kernel and a training set, we can form the matrix known as the kernel, or Gram matrix: the matrix containing the evaluation of the kernel function on all pairs of data points.

The study of patterns in data is as old as science. Consider, for example, the astronomical breakthroughs of Johannes Kepler formulated in his three famous laws of planetary motion. They can be viewed as relations that he detected in a large set of observational data compiled by Tycho Brahe.

Equally the wish to automate the search for patterns is at least as old as computing. The problem has been attacked using methods of statistics, machine learning, data mining and many other branches of science and engineering.

Pattern analysis deals with the problem of (automatically) detecting and characterising relations in data. Most statistical and machine learning methods of pattern analysis assume that the data is in vectorial form and that the relations can be expressed as classification rules, regression functions or cluster structures; these approaches often go under the general heading of ‘statistical pattern recognition’. ‘Syntactical’ or ‘structural pattern recognition’ represents an alternative approach that aims to detect rules among, for example, strings, often in the form of grammars or equivalent abstractions.

The evolution of automated algorithms for pattern analysis has undergone three revolutions. In the 1960s efficient algorithms for detecting linear relations within sets of vectors were introduced. Their computational and statistical behaviour was also analysed. The Perceptron algorithm introduced in 1957 is one example. The question of how to detect nonlinear relations was posed as a major research goal at that time.

As discussed in Chapter 1 perhaps the most important property of a pattern analysis algorithm is that it should identify statistically stable patterns. A stable relation is one that reflects some property of the source generating the data, and is therefore not a chance feature of the particular dataset. Proving that a given pattern is indeed significant is the concern of ‘learning theory’, a body of principles and methods that estimate the reliability of pattern functions under appropriate assumptions about the way in which the data was generated. The most common assumption is that the individual training examples are generated independently according to a fixed distribution, being the same distribution under which the expected value of the pattern function is small. Statistical analysis of the problem can therefore make use of the law of large numbers through the ‘concentration’ of certain random variables.

Concentration would be all that we need if we were only to consider one pattern function. Pattern analysis algorithms typically search for pattern functions over whole classes of functions, by choosing the function that best fits the particular training sample. We must therefore be able to prove stability not of a pre-defined pattern, but of one deliberately chosen for its fit to the data.

Clearly the more pattern functions at our disposal, the more likely that this choice could be a spurious pattern. The critical factor that controls how much our choice may have compromised the stability of the resulting pattern is the ‘capacity’ of the function class.