Computational capabilities of multilayer committee machines

We obtained an analytical expression for the computational complexity of many layered committee machines with a finite number of hidden layers (L < 8) using the generalization complexity measure introduced by Franco et al (2006) IEEE Trans. Neural Netw. 17 578. Although our result is valid in the large-size limit and for an overlap synaptic matrix that is ultrametric, it provides a useful tool for inferring the appropriate architecture a network must have to reproduce an arbitrary realizable Boolean function.

Advanced mean field methods:theory and practice

A major problem in modern probabilistic modeling is the huge computational complexity involved in typical calculations with multivariate probability distributions when the number of random variables is large. Because exact computations are infeasible in such cases and Monte Carlo sampling techniques may reach their limits, there is a need for methods that allow for efficient approximate computations. One of the simplest approximations is based on the mean field method, which has a long history in statistical physics. The method is widely used, particularly in the growing field of graphical models. Researchers from disciplines such as statistical physics, computer science, and mathematical statistics are studying ways to improve this and related methods and are exploring novel application areas. Leading approaches include the variational approach, which goes beyond factorizable distributions to achieve systematic improvements; the TAP (Thouless-Anderson-Palmer) approach, which incorporates correlations by including effective reaction terms in the mean field theory; and the more general methods of graphical models. Bringing together ideas and techniques from these diverse disciplines, this book covers the theoretical foundations of advanced mean field methods, explores the relation between the different approaches, examines the quality of the approximation obtained, and demonstrates their application to various areas of probabilistic modeling.

A basis function approach to Bayesian inference in diffusion processes

In this paper, we present a framework for Bayesian inference in continuous-time diffusion processes. The new method is directly related to the recently proposed variational Gaussian Process approximation (VGPA) approach to Bayesian smoothing of partially observed diffusions. By adopting a basis function expansion (BF-VGPA), both the time-dependent control parameters of the approximate GP process and its moment equations are projected onto a lower-dimensional subspace. This allows us both to reduce the computational complexity and to eliminate the time discretisation used in the previous algorithm. The new algorithm is tested on an Ornstein-Uhlenbeck process. Our preliminary results show that BF-VGPA algorithm provides a reasonably accurate state estimation using a small number of basis functions.

Parallel geostatistics for sparse and dense datasets

Very large spatially-referenced datasets, for example, those derived from satellite-based sensors which sample across the globe or large monitoring networks of individual sensors, are becoming increasingly common and more widely available for use in environmental decision making. In large or dense sensor networks, huge quantities of data can be collected over small time periods. In many applications the generation of maps, or predictions at specific locations, from the data in (near) real-time is crucial. Geostatistical operations such as interpolation are vital in this map-generation process and in emergency situations, the resulting predictions need to be available almost instantly, so that decision makers can make informed decisions and define risk and evacuation zones. It is also helpful when analysing data in less time critical applications, for example when interacting directly with the data for exploratory analysis, that the algorithms are responsive within a reasonable time frame. Performing geostatistical analysis on such large spatial datasets can present a number of problems, particularly in the case where maximum likelihood. Although the storage requirements only scale linearly with the number of observations in the dataset, the computational complexity in terms of memory and speed, scale quadratically and cubically respectively. Most modern commodity hardware has at least 2 processor cores if not more. Other mechanisms for allowing parallel computation such as Grid based systems are also becoming increasingly commonly available. However, currently there seems to be little interest in exploiting this extra processing power within the context of geostatistics. In this paper we review the existing parallel approaches for geostatistics. By recognising that diffeerent natural parallelisms exist and can be exploited depending on whether the dataset is sparsely or densely sampled with respect to the range of variation, we introduce two contrasting novel implementations of parallel algorithms based on approximating the data likelihood extending the methods of Vecchia [1988] and Tresp [2000]. Using parallel maximum likelihood variogram estimation and parallel prediction algorithms we show that computational time can be significantly reduced. We demonstrate this with both sparsely sampled data and densely sampled data on a variety of architectures ranging from the common dual core processor, found in many modern desktop computers, to large multi-node super computers. To highlight the strengths and weaknesses of the diffeerent methods we employ synthetic data sets and go on to show how the methods allow maximum likelihood based inference on the exhaustive Walker Lake data set.

Robust automatic mapping algorithms in a network monitoring scenario

Automatically generating maps of a measured variable of interest can be problematic. In this work we focus on the monitoring network context where observations are collected and reported by a network of sensors, and are then transformed into interpolated maps for use in decision making. Using traditional geostatistical methods, estimating the covariance structure of data collected in an emergency situation can be difficult. Variogram determination, whether by method-of-moment estimators or by maximum likelihood, is very sensitive to extreme values. Even when a monitoring network is in a routine mode of operation, sensors can sporadically malfunction and report extreme values. If this extreme data destabilises the model, causing the covariance structure of the observed data to be incorrectly estimated, the generated maps will be of little value, and the uncertainty estimates in particular will be misleading. Marchant and Lark [2007] propose a REML estimator for the covariance, which is shown to work on small data sets with a manual selection of the damping parameter in the robust likelihood. We show how this can be extended to allow treatment of large data sets together with an automated approach to all parameter estimation. The projected process kriging framework of Ingram et al. [2007] is extended to allow the use of robust likelihood functions, including the two component Gaussian and the Huber function. We show how our algorithm is further refined to reduce the computational complexity while at the same time minimising any loss of information. To show the benefits of this method, we use data collected from radiation monitoring networks across Europe. We compare our results to those obtained from traditional kriging methodologies and include comparisons with Box-Cox transformations of the data. We discuss the issue of whether to treat or ignore extreme values, making the distinction between the robust methods which ignore outliers and transformation methods which treat them as part of the (transformed) process. Using a case study, based on an extreme radiological events over a large area, we show how radiation data collected from monitoring networks can be analysed automatically and then used to generate reliable maps to inform decision making. We show the limitations of the methods and discuss potential extensions to remedy these.

Applications of fractals to image data compression

Digital image processing is exploited in many diverse applications but the size of digital images places excessive demands on current storage and transmission technology. Image data compression is required to permit further use of digital image processing. Conventional image compression techniques based on statistical analysis have reached a saturation level so it is necessary to explore more radical methods. This thesis is concerned with novel methods, based on the use of fractals, for achieving significant compression of image data within reasonable processing time without introducing excessive distortion. Images are modelled as fractal data and this model is exploited directly by compression schemes. The validity of this is demonstrated by showing that the fractal complexity measure of fractal dimension is an excellent predictor of image compressibility. A method of fractal waveform coding is developed which has low computational demands and performs better than conventional waveform coding methods such as PCM and DPCM. Fractal techniques based on the use of space-filling curves are developed as a mechanism for hierarchical application of conventional techniques. Two particular applications are highlighted: the re-ordering of data during image scanning and the mapping of multi-dimensional data to one dimension. It is shown that there are many possible space-filling curves which may be used to scan images and that selection of an optimum curve leads to significantly improved data compression. The multi-dimensional mapping property of space-filling curves is used to speed up substantially the lookup process in vector quantisation. Iterated function systems are compared with vector quantisers and the computational complexity or iterated function system encoding is also reduced by using the efficient matching algcnithms identified for vector quantisers.

Visual data mining: integrating machine learning with information visualization

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. Most existing systems concentrate either on mining algorithms or on visualization techniques. Though visual methods developed in information visualization have been helpful, for improved understanding of a complex large high-dimensional dataset, there is a need for an effective projection of such a dataset onto a lower-dimension (2D or 3D) manifold. This paper introduces a flexible visual data mining framework which combines advanced projection algorithms developed in the machine learning domain and visual techniques developed in the information visualization domain. The framework follows Shneiderman’s mantra to provide an effective user interface. The advantage of such an interface is that the user is directly involved in the data mining process. We integrate principled projection methods, such as Generative Topographic Mapping (GTM) and Hierarchical GTM (HGTM), with powerful visual techniques, such as magnification factors, directional curvatures, parallel coordinates, billboarding, and user interaction facilities, to provide an integrated visual data mining framework. Results on a real life high-dimensional dataset from the chemoinformatics domain are also reported and discussed. Projection results of GTM are analytically compared with the projection results from other traditional projection methods, and it is also shown that the HGTM algorithm provides additional value for large datasets. The computational complexity of these algorithms is discussed to demonstrate their suitability for the visual data mining framework.

Bayesian analysis of the scatterometer wind retrieval inverse problem:Some new approaches

The retrieval of wind vectors from satellite scatterometer observations is a non-linear inverse problem. A common approach to solving inverse problems is to adopt a Bayesian framework and to infer the posterior distribution of the parameters of interest given the observations by using a likelihood model relating the observations to the parameters, and a prior distribution over the parameters. We show how Gaussian process priors can be used efficiently with a variety of likelihood models, using local forward (observation) models and direct inverse models for the scatterometer. We present an enhanced Markov chain Monte Carlo method to sample from the resulting multimodal posterior distribution. We go on to show how the computational complexity of the inference can be controlled by using a sparse, sequential Bayes algorithm for estimation with Gaussian processes. This helps to overcome the most serious barrier to the use of probabilistic, Gaussian process methods in remote sensing inverse problems, which is the prohibitively large size of the data sets. We contrast the sampling results with the approximations that are found by using the sparse, sequential Bayes algorithm.

Pragmatic algorithms for implementing geostatistics with large datasets

With the ability to collect and store increasingly large datasets on modern computers comes the need to be able to process the data in a way that can be useful to a Geostatistician or application scientist. Although the storage requirements only scale linearly with the number of observations in the dataset, the computational complexity in terms of memory and speed, scale quadratically and cubically respectively for likelihood-based Geostatistics. Various methods have been proposed and are extensively used in an attempt to overcome these complexity issues. This thesis introduces a number of principled techniques for treating large datasets with an emphasis on three main areas: reduced complexity covariance matrices, sparsity in the covariance matrix and parallel algorithms for distributed computation. These techniques are presented individually, but it is also shown how they can be combined to produce techniques for further improving computational efficiency.

Force sensing for measuring human body movement

The research developed in this thesis explores the sensing and inference of human movement in a dynamic way, as opposed to conventional measurement systems, that are only concerned with discrete evaluations of stimuli in sequential time. Typically, conventional approaches are used to infer the dynamic movement of the body; such as vision and motion tracking devices, with either a human diagnosis or complex image processing algorithm to classify the movement. This research is therefore the first of its kind to attempt and provide a movement classifying algorithm through the use of minimal sensing points, with the application for this novel system, to classify human movement during a golf swing. There are two main categories of force sensing. Firstly, array-type systems consisting of many sensing elements, and are the most commonly researched and commercially available. Secondly, reduced force sensing element systems (RFSES) also known as distributive systems have only been recently exploited in the academic world. The fundamental difference between these systems is that array systems handle the data captured from each sensor as unique outputs and suffer the effects of resolution. The effect of resolution, is the error in the load position measurement between sensing elements, as the output is quantized in terms of position. This can be compared to a reduced sensor element system that maximises that data received through the coupling of data from a distribution of sensing points to describe the output in discrete time. Also this can be extended to a coupling of transients in the time domain to describe an activity or dynamic movement. It is the RFSES that is to be examined and exploited in the commercial sector due to its advantages over array-based approaches such as reduced design, computational complexity and cost.

Semantics of query-driven communication of exact values

We address the question of how to communicate among distributed processes valuessuch as real numbers, continuous functions and geometrical solids with arbitrary precision, yet efficiently. We extend the established concept of lazy communication using streams of approximants by introducing explicit queries. We formalise this approach using protocols of a query-answer nature. Such protocols enable processes to provide valid approximations with certain accuracy and focusing on certain locality as demanded by the receiving processes through queries. A lattice-theoretic denotational semantics of channel and process behaviour is developed. Thequery space is modelled as a continuous lattice in which the top element denotes the query demanding all the information, whereas other elements denote queries demanding partial and/or local information. Answers are interpreted as elements of lattices constructed over suitable domains of approximations to the exact objects. An unanswered query is treated as an error anddenoted using the top element. The major novel characteristic of our semantic model is that it reflects the dependency of answerson queries. This enables the definition and analysis of an appropriate concept of convergence rate, by assigning an effort indicator to each query and a measure of information content to eachanswer. Thus we capture not only what function a process computes, but also how a process transforms the convergence rates from its inputs to its outputs. In future work these indicatorscan be used to capture further computational complexity measures. A robust prototype implementation of our model is available.

Near-ML detection for MDL-impaired few-mode fiber transmission

Few-mode fiber transmission systems are typically impaired by mode-dependent loss (MDL). In an MDL-impaired link, maximum-likelihood (ML) detection yields a significant advantage in system performance compared to linear equalizers, such as zero-forcing and minimum-mean square error equalizers. However, the computational effort of the ML detection increases exponentially with the number of modes and the cardinality of the constellation. We present two methods that allow for near-ML performance without being afflicted with the enormous computational complexity of ML detection: improved reduced-search ML detection and sphere decoding. Both algorithms are tested regarding their performance and computational complexity in simulations of three and six spatial modes with QPSK and 16QAM constellations.

Nonlinearity compensation in multi-rate 28 Gbaud WDM systems employing optical and digital techniques under diverse link configurations

Digital back-propagation (DBP) has recently been proposed for the comprehensive compensation of channel nonlinearities in optical communication systems. While DBP is attractive for its flexibility and performance, it poses significant challenges in terms of computational complexity. Alternatively, phase conjugation or spectral inversion has previously been employed to mitigate nonlinear fibre impairments. Though spectral inversion is relatively straightforward to implement in optical or electrical domain, it requires precise positioning and symmetrised link power profile in order to avail the full benefit. In this paper, we directly compare ideal and low-precision single-channel DBP with single-channel spectral-inversion both with and without symmetry correction via dispersive chirping. We demonstrate that for all the dispersion maps studied, spectral inversion approaches the performance of ideal DBP with 40 steps per span and exceeds the performance of electronic dispersion compensation by ~3.5 dB in Q-factor, enabling up to 96% reduction in complexity in terms of required DBP stages, relative to low precision one step per span based DBP. For maps where quasi-phase matching is a significant issue, spectral inversion significantly outperforms ideal DBP by ~3 dB.

Colour image coding with wavelets and matching pursuit

This thesis considers sparse approximation of still images as the basis of a lossy compression system. The Matching Pursuit (MP) algorithm is presented as a method particularly suited for application in lossy scalable image coding. Its multichannel extension, capable of exploiting inter-channel correlations, is found to be an efficient way to represent colour data in RGB colour space. Known problems with MP, high computational complexity of encoding and dictionary design, are tackled by finding an appropriate partitioning of an image. The idea of performing MP in the spatio-frequency domain after transform such as Discrete Wavelet Transform (DWT) is explored. The main challenge, though, is to encode the image representation obtained after MP into a bit-stream. Novel approaches for encoding the atomic decomposition of a signal and colour amplitudes quantisation are proposed and evaluated. The image codec that has been built is capable of competing with scalable coders such as JPEG 2000 and SPIHT in terms of compression ratio.

Visual data mining using principled projection algorithms and information visualization techniques

We introduce a flexible visual data mining framework which combines advanced projection algorithms from the machine learning domain and visual techniques developed in the information visualization domain. The advantage of such an interface is that the user is directly involved in the data mining process. We integrate principled projection algorithms, such as generative topographic mapping (GTM) and hierarchical GTM (HGTM), with powerful visual techniques, such as magnification factors, directional curvatures, parallel coordinates and billboarding, to provide a visual data mining framework. Results on a real-life chemoinformatics dataset using GTM are promising and have been analytically compared with the results from the traditional projection methods. It is also shown that the HGTM algorithm provides additional value for large datasets. The computational complexity of these algorithms is discussed to demonstrate their suitability for the visual data mining framework. Copyright 2006 ACM.