The generalization performance of some of the learning rules introduced in the previous chapter could be characterized either by using simple arguments from statistics as in the case of the Hebb rule, or by exploiting our results on Gibbs learning obtained in chapter 2 as in the case of the Bayes rule. Neither of these attempts is successful, however, in determining the generalization error of the remaining learning rules.

In this chapter we will introduce several modifications of the central statistical mechanics method introduced in chapter 2 which will allow us to analyse the generalization behaviour of these remaining rules. The main observation is that all these learning rules can be interpreted as prescriptions to minimize appropriately chosen cost functions. Generalizing the concept of Gibbs learning to non-zero training error will pave the way to studying such minimization problems in a unified fashion.

Before embarking on these general considerations, however, we will discuss in the first section of this chapter how learning rules aiming at maximal stabilities are most conveniently analysed.

The main results of this chapter concerning the generalization error of the various rules are summarized in fig. 4.3 and table 4.1.

Maximal stabilities

A minor extension of the statistical mechanics formalism introduced in chapter 2 is sufficient to analyse the generalization performance of the adatron and the pseudo-inverse rule. The common feature of these two rules is that they search for couplings with maximal stabilities, formalized by the maximization of the stability parameter K.

So far we have focused on the performance of various learning rules as a function of the size of the training set with examples which are all selected before training starts and remain available during the whole training period. However, in both real life and many practical situations, the training examples come and go with time. Learning then has to proceed on-line, using only the training example which is available at any particular time. This is to be contrasted with the previous scenario, called off-line or batch learning, in which all the training examples are available at all times.

For the Hebb rule, the off-line and on-line scenario coincide: each example provides an additive contribution to the synaptic vector, which is independent of the other examples. We mentioned already in chapter 3 that this rule performs rather badly for large training sets, precisely because it treats all the learning examples in exactly the same way. The purpose of this chapter is to introduce more advanced or alternative on-line learning rules, and to compare their performance with that of their off-line versions.

Stochastic gradient descent

In an on-line scenario, the training examples are presented once and in a sequential order and the coupling vector J is updated at each time step using information from this single example only.

Introduction

In this chapter we investigate ICA models in which the number of sources, M, may be less than the number of sensors, N: so-called non-square mixing.

The ‘extra’ sensor observations are explained as observation noise. This general approach may be called Probabilistic Independent Component Analysis (PICA) by analogy with the Probabilistic Principal Component Analysis (PPCA) model of Tipping & Bishop [I9971; ICA and PCA don't have observation noise, PICA and PPCA do.

Non-square ICA models give rise to a likelihood model for the data involving an integral which is intractable. In this chapter we build on previous work in which the integral is estimated using a Laplace approximation. By making the further assumption that the unmixing matrix lies on the decorrelating manifold we are able to make a number of simplifications. Firstly, the observation noise can be estimated using PCA methods, and, secondly, optimisation takes place in a space having a much reduced dimensionality, having order M2 parameters rather than M × N. Again, building on previous work, we derive a model order selection criterion for selecting the appropriate number of sources. This is based on the Laplace approximation as applied to the decorrelating manifold. This is then compared with PCA model order selection methods on music and EEG datasets.

Non-Gaussianity is of paramount importance in ICA estimation. Without non-Gaussianity the estimation is not possible at all (unless the independent components have time-dependences). Therefore, it is not surprising that non-Gaussianity could be used as a leading principle in ICA estimation.

In this chapter, we derive a simple principle of ICA estimation: the independent components can be found as the projections that maximize non-Gaussianity. In addition to its intuitive appeal, this approach allows us to derive a highly efficient ICA algorithm, Fast ICA. This is a fixed-point algorithm that can be used for estimating the independent components one by one. At the end of the chapter, it will be seen that it is closely connected to maximum likelihood or infomax estimation as well.

Whitening

First, let us consider preprocessing techniques that are essential if we want to develop fast ICA methods.

The rather trivial preprocessing that is used in many cases is to centre x, i.e. subtract its mean vector m = ε{x} so as to make x a zero-mean variable. This implies that s is zero-mean as well. This preprocessing is made solely to simplify the ICA algorithms: it does not mean that the mean could not be estimated. After estimating the mixing matrix A with centred data, we can complete the estimation by adding the mean vector of s back to the centred estimates of s.

An unsupervised classification algorithm is derived by modelling observed data as a mixture of several mutually exclusive classes that are each described by linear combinations of independent, non-Gaussian densities. The algorithm estimates the density of each class and is able to model class distributions with non-Gaussian structure. It can improve classification accuracy compared with standard Gaussian mixture models. When applied to images, the algorithm can learn efficient codes (basis functions) for images that capture the statistical structure of the images. We applied this method to the problem of unsupervised classification, segmentation and de-noising of images. This method was effective in classifying complex image textures such as trees and rocks in natural scenes. It was also useful for de-noising and filling in missing pixels in images with complex structures. The advantage of this model is that image codes can be learned with increasing numbers of classes thus providing greater flexibility in modelling structure and in finding more image features than in either Gaussian mixture models or standard ICA algorithms.

Introduction

Recently, Blind Source Separation by Independent Component Analysis has been applied to signal processing problems including speech enhancement, telecommunications and medical signal processing. ICA finds a linear non-orthogonal coordinate system in multivariate data determined by second- and higher-order statistics. The goal of ICA is to linearly transform the data in such a way that the transformed variables are as statistically independent from each other as possible [Jutten & Herault, 1991, Comon, 1994, Bell & Sejnowski, 1995, Cardoso & Laheld, 1996, Lee et al., 2000b]. ICA generalizes the technique of Principal Component Analysis (PCA) and, like PCA, has proven a useful tool for finding structure in data.

Introduction

Independent Component Analysis (ICA), as a signal processing tool, has shown great promise in many application domains, two of the most successful being telecommunications and biomedical engineering. With the growing awareness of ICA many other less obvious applications of the transform are starting to appear, for example financial time series prediction [Back & Weigend, 19981 and information retrieval [Isbell & Viola, 1999, Girolami, 2000a, Vinokourov & Girolami, 20001. In such applications ICA is being used as an unsupervised means of exploring and, hopefully, uncovering meaningful latent traits or structure within the data. In the case of financial time series prediction the factors which drive the evolution of the time series are hopefully uncovered, whereas within information retrieval the latent concepts or topics which generate key-words that occur in the documents are sought after. The strong assumption of independence of the hidden factors in ICA is difficult to argue for when there is limited a priori knowledge of the data, indeed it is desired that the analysis uncover injormative components which may, or may not, be independent. Nevertheless the independence assumption allows analytically tractable statistical models to be developed.

This chapter will consider how the standard ICA model can be extended and used in the unsupervised classification and visualisation of multivariate data. Prior to the formal presentation, to set the context of the remainder of the chapter, a short review of the ICA signal model and the corresponding statistical representation is given. The remaining sections propose ICA inspired techniques for the unsupervised classification and visualisation of multivariate data.

In recent years there has been an explosion of interest in the application and theory of independent component analysis (ICA). This book is aimed to provide a self-contained introduction to the subject as well as offering a set of invited contributions which we see as lying at the cutting edge of ICA research.

ICA is intimately linked with the problem of blind source separation-attempting to recover a set of underlying sources when only a noisy mapping from these sources, the observations, is given-and we regard this as the canonical form of ICA. Until recently this mapping was taken to be linear (but see Chapter 4) and “traditionally” (if tradition is allowed in a field of such recent developments) noiseless with the number of observations being equal to the number of hypothesised sources. It is surprising that even the simplest of ICA models can be invaluable and offer new insights into data analysis and interpretation. This, at first sight unreasonable, claim may be supported by noting that many observations of physical systems are produced by a linear combination of underlying sources. Furthermore, in many applications, it is an end in itself to produce a set of “sources” which are statistically independent rather than just decorrelated (see Chapter 1) and for this ICA would appear an ideal tool.

Introduction

One may think of blind source separation as the problem of identifying speakers (sources) in a room given only recordings from a number of microphones, each of which records a linear mixture of the sources, whose statistical characteristics are unknown.

Here we consider the blind source separation problem when the mixing of the sources is non-stationary. Pursuing the speakers in a room analogy, we address the problem of identifying the speakers when they (or equivalently, the microphones) are moving. The problem is cast in terms of a hidden state (the mixing proportions of the sources) which we track using particle filter methods, which permit the tracking of arbitrary state densities. Murata et al. [I9971 have addressed this problem by adapting the learning rate and we mention work by Penny et al. [2000] on hidden Markov models for ICA which allows for abrupt changes in the mixing matrix with stationary periods in between.

We first briefly re-review classical Independent Component Analysis. ICA with non-stationary mixing is described in terms of a hidden state model and methods for estimating the sources and the mixing are described. Particle filter techniques are then introduced for the modelling of state densities. Finally, we address the non-stationary mixing problem when the sources are independent, but possess temporal correlations.

This chapter deals with independent component analysis and blind source separation for nonlinear data models. A fundamental difficulty, especially in the nonlinear ICA problem, is that it is highly non-unique without a suitable regularization. After considering this, two methods for solving the nonlinear ICA and BSS problems are presented in more detail. The first one is a maximum likelihood method based on a modified generative topographic mapping. The second approach applies Bayesian ensemble learning to a flexible multi-layer perceptron model for finding the sources and nonlinear mixing mapping that have most probably given rise to the observed mixed data. Finally, other techniques introduced for the nonlinear ICA and BSS problems are briefly reviewed.

Introduction

Independent Component Analysis [Lee, 1998, Oja et al., 1997, Girolami, 1999bl is a statistical technique which tries to represent the observed data in terms of statistically independent component variables. ICA is closely related to the blind source separation (BSS) problem [Cardoso, 1998a, Amari et al., 1996, Lee, 1998, Oja et al., 1997, Girolami, 1999b1, where the general goal is to separate mutually independent but otherwise unknown source signals from their observed mixtures without knowing the mixing process.

Introduction

Independent Component Analysis (ICA) has recently become an important tool for modelling and understanding empirical datasets as it offers an elegant and practical methodology for blind source separation and deconvolution. It is seldom possible to observe a pure unadulterated signal. Instead most observations consist of a mixture of signals usually corrupted by noise, and frequently filtered. The signal processing community has devoted much attention to the problem of recovering the constituent sources from the convolutive mixture; ICA may be applied to this Blind Source Separation (BSS) problem to recover the sources. As the appellation independent suggests, recovery relies on the assumption that the constituent sources are mutually independent.

Finding a natural coordinate system is an essential first step in the analysis of empirical data. Principal component analysis (PCA) has, for many years, been used to find a set of basis vectors which are determined by the dataset itself. The principal components are orthogonal and projections of the data onto them are linearly decorrelated, properties which can be ensured by considering only the second order statistical characteristics of the data. ICA aims at a loftier goal: it seeks a transformation to coordinates in which the data are maximally statistically independent, not merely decorrelated.

This chapter shows how to take advantage of the non-stationarity of the source signals to achieve the blind separation of instantaneous mixtures. Off-line and on-line algorithms are developed based on the likelihood principle and the mutual information objective. The analysis of the achievable performance also reveals how non-stationarity and non-Gaussianity both contribute to the separability.

Introduction

This chapter addresses the problem of separating instantaneous mixtures of non-stationary sources. More to the point, we investigate how to take advantage of a possible non-stationarity of the sources in order to achieve separation.

When the source signals are modelled as i.i.d. (independently and identically distributed) sequences, blind separation can be achieved only by exploiting the non-Gaussianity of the sources. In this case, sources must be non-Gaussian to be separated; poor results are to be expected when the (marginal) distributions of source signals are too close to a Gaussian distribution. In contrast, the existence of a temporal structure in the source signals makes it possible to separate even Gaussian sources. Two types of temporal structures have been considered in the literature, each corresponding to breaking one ‘i’ in i.i.d. The first case is that of stationary signals with a temporal dependence, thus having a non-flat spectrum; these are ‘non-white’ or ‘coloured’ sequences. Identifiability is granted provided no two sources have proportional spectra (see e.g., [Pham & Garrat, 19931 or [Belouchrani et al., 19971 for more elaborate statements and some algorithms) even when the signals are normally distributed.

Most approaches to the problem of source separation use the assumption of statistical independence. To capture statistical independence higher order statistics are required. In this chapter we will demonstrate how higher order criteria, such as maximum kurtosis, arise naturally from the property of non-stationarity. We will also show that source separation of non-stationary signals can be based entirely on second order statistics of the signals. Natural signals, be they images or time sequences, are for the most part non-stationary. For natural signals therefore we argue that non-stationarity is the fundamental property, from which specific second or higher order separation criteria can be derived. We contrast the linear bases obtained using second order non-stationarity and ICA for the cases of natural images and speech powers. Based on these results we argue that speech powers can in fact be understood as a linear superposition of non-stationary spectro-temporal independent components, while this is not so evident for a spatial basis of images intensities. Finally we demonstrate the practical utility of the second order non-stationarity concept with a separation algorithm for the problem of convolutive source separation. We show its effectiveness on acoustic mixtures in real reverberant environments.

Second and higher order separation criteria in the context of non-stationary signals

Most approaches to source separation have been based on the condition of statistical independence of the constituent signals. Conventionally, higher order statistics are required to capture statistical independence. In fact if the source signals are identically and independently distributed (i.i.d.) samples of a stationary distribution second order statistics are not sufficient for Separation. Fortunately, natural signals are often not stationary, or independently distributed.