Incorporating curvature information into on-line learning

We analyse the dynamics of a number of second order on-line learning algorithms training multi-layer neural networks, using the methods of statistical mechanics. We first consider on-line Newton's method, which is known to provide optimal asymptotic performance. We determine the asymptotic generalization error decay for a soft committee machine, which is shown to compare favourably with the result for standard gradient descent. Matrix momentum provides a practical approximation to this method by allowing an efficient inversion of the Hessian. We consider an idealized matrix momentum algorithm which requires access to the Hessian and find close correspondence with the dynamics of on-line Newton's method. In practice, the Hessian will not be known on-line and we therefore consider matrix momentum using a single example approximation to the Hessian. In this case good asymptotic performance may still be achieved, but the algorithm is now sensitive to parameter choice because of noise in the Hessian estimate. On-line Newton's method is not appropriate during the transient learning phase, since a suboptimal unstable fixed point of the gradient descent dynamics becomes stable for this algorithm. A principled alternative is to use Amari's natural gradient learning algorithm and we show how this method provides a significant reduction in learning time when compared to gradient descent, while retaining the asymptotic performance of on-line Newton's method.

Structured neural network modelling of multi-valued functions for wind retrieval from scatterometer measurements

A conventional neural network approach to regression problems approximates the conditional mean of the output vector. For mappings which are multi-valued this approach breaks down, since the average of two solutions is not necessarily a valid solution. In this article mixture density networks, a principled method to model conditional probability density functions, are applied to retrieving Cartesian wind vector components from satellite scatterometer data. A hybrid mixture density network is implemented to incorporate prior knowledge of the predominantly bimodal function branches. An advantage of a fully probabilistic model is that more sophisticated and principled methods can be used to resolve ambiguities.

Bayesian inference for wind field retrieval

In many problems in spatial statistics it is necessary to infer a global problem solution by combining local models. A principled approach to this problem is to develop a global probabilistic model for the relationships between local variables and to use this as the prior in a Bayesian inference procedure. We show how a Gaussian process with hyper-parameters estimated from Numerical Weather Prediction Models yields meteorologically convincing wind fields. We use neural networks to make local estimates of wind vector probabilities. The resulting inference problem cannot be solved analytically, but Markov Chain Monte Carlo methods allow us to retrieve accurate wind fields.

GTM: the generative topographic mapping

This thesis describes the Generative Topographic Mapping (GTM) --- a non-linear latent variable model, intended for modelling continuous, intrinsically low-dimensional probability distributions, embedded in high-dimensional spaces. It can be seen as a non-linear form of principal component analysis or factor analysis. It also provides a principled alternative to the self-organizing map --- a widely established neural network model for unsupervised learning --- resolving many of its associated theoretical problems. An important, potential application of the GTM is visualization of high-dimensional data. Since the GTM is non-linear, the relationship between data and its visual representation may be far from trivial, but a better understanding of this relationship can be gained by computing the so-called magnification factor. In essence, the magnification factor relates the distances between data points, as they appear when visualized, to the actual distances between those data points. There are two principal limitations of the basic GTM model. The computational effort required will grow exponentially with the intrinsic dimensionality of the density model. However, if the intended application is visualization, this will typically not be a problem. The other limitation is the inherent structure of the GTM, which makes it most suitable for modelling moderately curved probability distributions of approximately rectangular shape. When the target distribution is very different to that, theaim of maintaining an `interpretable' structure, suitable for visualizing data, may come in conflict with the aim of providing a good density model. The fact that the GTM is a probabilistic model means that results from probability theory and statistics can be used to address problems such as model complexity. Furthermore, this framework provides solid ground for extending the GTM to wider contexts than that of this thesis.

Regularisation of mixture density networks

Mixture Density Networks are a principled method to model conditional probability density functions which are non-Gaussian. This is achieved by modelling the conditional distribution for each pattern with a Gaussian Mixture Model for which the parameters are generated by a neural network. This thesis presents a novel method to introduce regularisation in this context for the special case where the mean and variance of the spherical Gaussian Kernels in the mixtures are fixed to predetermined values. Guidelines for how these parameters can be initialised are given, and it is shown how to apply the evidence framework to mixture density networks to achieve regularisation. This also provides an objective stopping criteria that can replace the `early stopping' methods that have previously been used. If the neural network used is an RBF network with fixed centres this opens up new opportunities for improved initialisation of the network weights, which are exploited to start training relatively close to the optimum. The new method is demonstrated on two data sets. The first is a simple synthetic data set while the second is a real life data set, namely satellite scatterometer data used to infer the wind speed and wind direction near the ocean surface. For both data sets the regularisation method performs well in comparison with earlier published results. Ideas on how the constraint on the kernels may be relaxed to allow fully adaptable kernels are presented.

First year qualifying report: neural networks for extracting wind vectors from satellite scatterometer data

The ERS-1 Satellite was launched in July 1991 by the European Space Agency into a polar orbit at about km800, carrying a C-band scatterometer. A scatterometer measures the amount of radar back scatter generated by small ripples on the ocean surface induced by instantaneous local winds. Operational methods that extract wind vectors from satellite scatterometer data are based on the local inversion of a forward model, mapping scatterometer observations to wind vectors, by the minimisation of a cost function in the scatterometer measurement space.par This report uses mixture density networks, a principled method for modelling conditional probability density functions, to model the joint probability distribution of the wind vectors given the satellite scatterometer measurements in a single cell (the `inverse' problem). The complexity of the mapping and the structure of the conditional probability density function are investigated by varying the number of units in the hidden layer of the multi-layer perceptron and the number of kernels in the Gaussian mixture model of the mixture density network respectively. The optimal model for networks trained per trace has twenty hidden units and four kernels. Further investigation shows that models trained with incidence angle as an input have results comparable to those models trained by trace. A hybrid mixture density network that incorporates geophysical knowledge of the problem confirms other results that the conditional probability distribution is dominantly bimodal.par The wind retrieval results improve on previous work at Aston, but do not match other neural network techniques that use spatial information in the inputs, which is to be expected given the ambiguity of the inverse problem. Current work uses the local inverse model for autonomous ambiguity removal in a principled Bayesian framework. Future directions in which these models may be improved are given.

Natural gradient matrix momentum

Natural gradient learning is an efficient and principled method for improving on-line learning. In practical applications there will be an increased cost required in estimating and inverting the Fisher information matrix. We propose to use the matrix momentum algorithm in order to carry out efficient inversion and study the efficacy of a single step estimation of the Fisher information matrix. We analyse the proposed algorithm in a two-layer network, using a statistical mechanics framework which allows us to describe analytically the learning dynamics, and compare performance with true natural gradient learning and standard gradient descent.

On-line learning in neural networks

On-line learning is one of the most powerful and commonly used techniques for training large layered networks and has been used successfully in many real-world applications. Traditional analytical methods have been recently complemented by ones from statistical physics and Bayesian statistics. This powerful combination of analytical methods provides more insight and deeper understanding of existing algorithms and leads to novel and principled proposals for their improvement. This book presents a coherent picture of the state-of-the-art in the theoretical analysis of on-line learning. An introduction relates the subject to other developments in neural networks and explains the overall picture. Surveys by leading experts in the field combine new and established material and enable non-experts to learn more about the techniques and methods used. This book, the first in the area, provides a comprehensive view of the subject and will be welcomed by mathematicians, scientists and engineers, whether in industry or academia.

Semi-supervised construction of general visualization hierarchies

We have recently developed a principled approach to interactive non-linear hierarchical visualization [8] based on the Generative Topographic Mapping (GTM). Hierarchical plots are needed when a single visualization plot is not sufficient (e.g. when dealing with large quantities of data). In this paper we extend our system by giving the user a choice of initializing the child plots of the current plot in either interactive, or automatic mode. In the interactive mode the user interactively selects ``regions of interest'' as in [8], whereas in the automatic mode an unsupervised minimum message length (MML)-driven construction of a mixture of GTMs is used. The latter is particularly useful when the plots are covered with dense clusters of highly overlapping data projections, making it difficult to use the interactive mode. Such a situation often arises when visualizing large data sets. We illustrate our approach on a data set of 2300 18-dimensional points and mention extension of our system to accommodate discrete data types.

DVMS 1.5: A user manual (the data visualization and modeling system)

The data available during the drug discovery process is vast in amount and diverse in nature. To gain useful information from such data, an effective visualisation tool is required. To provide better visualisation facilities to the domain experts (screening scientist, biologist, chemist, etc.),we developed a software which is based on recently developed principled visualisation algorithms such as Generative Topographic Mapping (GTM) and Hierarchical Generative Topographic Mapping (HGTM). The software also supports conventional visualisation techniques such as Principal Component Analysis, NeuroScale, PhiVis, and Locally Linear Embedding (LLE). The software also provides global and local regression facilities . It supports regression algorithms such as Multilayer Perceptron (MLP), Radial Basis Functions network (RBF), Generalised Linear Models (GLM), Mixture of Experts (MoE), and newly developed Guided Mixture of Experts (GME). This user manual gives an overview of the purpose of the software tool, highlights some of the issues to be taken care while creating a new model, and provides information about how to install & use the tool. The user manual does not require the readers to have familiarity with the algorithms it implements. Basic computing skills are enough to operate the software.

Independent component analysis for domain independent watermarking

A new principled domain independent watermarking framework is presented. The new approach is based on embedding the message in statistically independent sources of the covertext to mimimise covertext distortion, maximise the information embedding rate and improve the method's robustness against various attacks. Experiments comparing the performance of the new approach, on several standard attacks show the current proposed approach to be competitive with other state of the art domain-specific methods.

ICA for watermarking digital images

A domain independent ICA-based approach to watermarking is presented. This approach can be used on images, music or video to embed either a robust or fragile watermark. In the case of robust watermarking, the method shows high information rate and robustness against malicious and non-malicious attacks, while keeping a low induced distortion. The fragile watermarking scheme, on the other hand, shows high sensitivity to tampering attempts while keeping the requirement for high information rate and low distortion. The improved performance is achieved by employing a set of statistically independent sources (the independent components) as the feature space and principled statistical decoding methods. The performance of the suggested method is compared to other state of the art approaches. The paper focuses on applying the method to digitized images although the same approach can be used for other media, such as music or video.

Multi-valued control problems and mixture density network

We have proposed a novel robust inversion-based neurocontroller that searches for the optimal control law by sampling from the estimated Gaussian distribution of the inverse plant model. However, for problems involving the prediction of continuous variables, a Gaussian model approximation provides only a very limited description of the properties of the inverse model. This is usually the case for problems in which the mapping to be learned is multi-valued or involves hysteritic transfer characteristics. This often arises in the solution of inverse plant models. In order to obtain a complete description of the inverse model, a more general multicomponent distributions must be modeled. In this paper we test whether our proposed sampling approach can be used when considering an arbitrary conditional probability distributions. These arbitrary distributions will be modeled by a mixture density network. Importance sampling provides a structured and principled approach to constrain the complexity of the search space for the ideal control law. The effectiveness of the importance sampling from an arbitrary conditional probability distribution will be demonstrated using a simple single input single output static nonlinear system with hysteretic characteristics in the inverse plant model.

Robust control of nonlinear stochastic systems by modelling conditional distributions of control signals

We introduce a novel inversion-based neuro-controller for solving control problems involving uncertain nonlinear systems that could also compensate for multi-valued systems. The approach uses recent developments in neural networks, especially in the context of modelling statistical distributions, which are applied to forward and inverse plant models. Provided that certain conditions are met, an estimate of the intrinsic uncertainty for the outputs of neural networks can be obtained using the statistical properties of networks. More generally, multicomponent distributions can be modelled by the mixture density network. In this work a novel robust inverse control approach is obtained based on importance sampling from these distributions. This importance sampling provides a structured and principled approach to constrain the complexity of the search space for the ideal control law. The performance of the new algorithm is illustrated through simulations with example systems.

miniDVMS v1.8 : A user manual(The data visualization and modeling system)

Today, the data available to tackle many scientific challenges is vast in quantity and diverse in nature. The exploration of heterogeneous information spaces requires suitable mining algorithms as well as effective visual interfaces. miniDVMS v1.8 provides a flexible visual data mining framework which combines advanced projection algorithms developed in the machine learning domain and visual techniques developed in the information visualisation domain. The advantage of this interface is that the user is directly involved in the data mining process. Principled projection methods, such as generative topographic mapping (GTM) and hierarchical GTM (HGTM), are integrated with powerful visual techniques, such as magnification factors, directional curvatures, parallel coordinates, and user interaction facilities, to provide this integrated visual data mining framework. The software also supports conventional visualisation techniques such as principal component analysis (PCA), Neuroscale, and PhiVis. This user manual gives an overview of the purpose of the software tool, highlights some of the issues to be taken care while creating a new model, and provides information about how to install and use the tool. The user manual does not require the readers to have familiarity with the algorithms it implements. Basic computing skills are enough to operate the software.