Sparse bayesian learning and the relevance vector machine
TL;DR: It is demonstrated that by exploiting a probabilistic Bayesian learning framework, the 'relevance vector machine' (RVM) can derive accurate prediction models which typically utilise dramatically fewer basis functions than a comparable SVM while offering a number of additional advantages.
Abstract: This paper introduces a general Bayesian framework for obtaining sparse solutions to regression and classification tasks utilising models linear in the parameters Although this framework is fully general, we illustrate our approach with a particular specialisation that we denote the 'relevance vector machine' (RVM), a model of identical functional form to the popular and state-of-the-art 'support vector machine' (SVM) We demonstrate that by exploiting a probabilistic Bayesian learning framework, we can derive accurate prediction models which typically utilise dramatically fewer basis functions than a comparable SVM while offering a number of additional advantages These include the benefits of probabilistic predictions, automatic estimation of 'nuisance' parameters, and the facility to utilise arbitrary basis functions (eg non-'Mercer' kernels) We detail the Bayesian framework and associated learning algorithm for the RVM, and give some illustrative examples of its application along with some comparative benchmarks We offer some explanation for the exceptional degree of sparsity obtained, and discuss and demonstrate some of the advantageous features, and potential extensions, of Bayesian relevance learning
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23 Nov 2005TL;DR: The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics, and deals with the supervised learning problem for both regression and classification.
Abstract: A comprehensive and self-contained introduction to Gaussian processes, which provide a principled, practical, probabilistic approach to learning in kernel machines. Gaussian processes (GPs) provide a principled, practical, probabilistic approach to learning in kernel machines. GPs have received increased attention in the machine-learning community over the past decade, and this book provides a long-needed systematic and unified treatment of theoretical and practical aspects of GPs in machine learning. The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics. The book deals with the supervised-learning problem for both regression and classification, and includes detailed algorithms. A wide variety of covariance (kernel) functions are presented and their properties discussed. Model selection is discussed both from a Bayesian and a classical perspective. Many connections to other well-known techniques from machine learning and statistics are discussed, including support-vector machines, neural networks, splines, regularization networks, relevance vector machines and others. Theoretical issues including learning curves and the PAC-Bayesian framework are treated, and several approximation methods for learning with large datasets are discussed. The book contains illustrative examples and exercises, and code and datasets are available on the Web. Appendixes provide mathematical background and a discussion of Gaussian Markov processes.
11,357 citations
Cites background or methods from "Sparse bayesian learning and the re..."
...relevance vectors Empirically it is often observed that the number of relevance vectors is smaller than the number of support vectors on the same problem [Tipping, 2001]....
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...In this section we present a different kind of analysis within the probably approximately correct (PAC) PAC framework due to Valiant [1984]. Seeger [2002; 2003] has presented a PACBayesian analysis of generalization in Gaussian process classifiers and we get to this in a number of stages; we first present an introduction to the PAC framework (section 7....
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...Although usually not presented as such, the relevance vector machine (RVM) introduced by Tipping [2001] is actually a special case of a Gaussian process....
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...The original RVM algorithm [Tipping, 2001] was not able to exploit the sparsity very effectively during model fitting as it was initialized with all of the αis set to finite values, meaning that all of the basis functions contributed to the model....
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TL;DR: DARTEL has been applied to intersubject registration of 471 whole brain images, and the resulting deformations were evaluated in terms of how well they encode the shape information necessary to separate male and female subjects and to predict the ages of the subjects.
6,999 citations
TL;DR: The goal of this article is to review the state-of-the-art tracking methods, classify them into different categories, and identify new trends to discuss the important issues related to tracking including the use of appropriate image features, selection of motion models, and detection of objects.
Abstract: The goal of this article is to review the state-of-the-art tracking methods, classify them into different categories, and identify new trends. Object tracking, in general, is a challenging problem. Difficulties in tracking objects can arise due to abrupt object motion, changing appearance patterns of both the object and the scene, nonrigid object structures, object-to-object and object-to-scene occlusions, and camera motion. Tracking is usually performed in the context of higher-level applications that require the location and/or shape of the object in every frame. Typically, assumptions are made to constrain the tracking problem in the context of a particular application. In this survey, we categorize the tracking methods on the basis of the object and motion representations used, provide detailed descriptions of representative methods in each category, and examine their pros and cons. Moreover, we discuss the important issues related to tracking including the use of appropriate image features, selection of motion models, and detection of objects.
5,318 citations
TL;DR: This Review looks at some key brain theories in the biological and physical sciences from the free-energy perspective, suggesting that several global brain theories might be unified within a free- energy framework.
Abstract: A free-energy principle has been proposed recently that accounts for action, perception and learning. This Review looks at some key brain theories in the biological (for example, neural Darwinism) and physical (for example, information theory and optimal control theory) sciences from the free-energy perspective. Crucially, one key theme runs through each of these theories — optimization. Furthermore, if we look closely at what is optimized, the same quantity keeps emerging, namely value (expected reward, expected utility) or its complement, surprise (prediction error, expected cost). This is the quantity that is optimized under the free-energy principle, which suggests that several global brain theories might be unified within a free-energy framework.
4,866 citations
TL;DR: For instance, mean-field variational inference as discussed by the authors approximates probability densities through optimization, which is used in many applications and tends to be faster than classical methods, such as Markov chain Monte Carlo sampling.
Abstract: One of the core problems of modern statistics is to approximate difficult-to-compute probability densities. This problem is especially important in Bayesian statistics, which frames all inference about unknown quantities as a calculation involving the posterior density. In this article, we review variational inference (VI), a method from machine learning that approximates probability densities through optimization. VI has been used in many applications and tends to be faster than classical methods, such as Markov chain Monte Carlo sampling. The idea behind VI is to first posit a family of densities and then to find a member of that family which is close to the target density. Closeness is measured by Kullback–Leibler divergence. We review the ideas behind mean-field variational inference, discuss the special case of VI applied to exponential family models, present a full example with a Bayesian mixture of Gaussians, and derive a variant that uses stochastic optimization to scale up to massive data...
3,421 citations
References
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01 Jan 1995TL;DR: Setting of the learning problem consistency of learning processes bounds on the rate of convergence ofLearning processes controlling the generalization ability of learning process constructing learning algorithms what is important in learning theory?
Abstract: Setting of the learning problem consistency of learning processes bounds on the rate of convergence of learning processes controlling the generalization ability of learning processes constructing learning algorithms what is important in learning theory?.
40,147 citations
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TL;DR: Presenting a method for determining the necessary and sufficient conditions for consistency of learning process, the author covers function estimates from small data pools, applying these estimations to real-life problems, and much more.
Abstract: A comprehensive look at learning and generalization theory. The statistical theory of learning and generalization concerns the problem of choosing desired functions on the basis of empirical data. Highly applicable to a variety of computer science and robotics fields, this book offers lucid coverage of the theory as a whole. Presenting a method for determining the necessary and sufficient conditions for consistency of learning process, the author covers function estimates from small data pools, applying these estimations to real-life problems, and much more.
26,531 citations
"Sparse bayesian learning and the re..." refers background or methods in this paper
...The function sinc(x) = sin(x)=x has been a popular choice to illustrate support vector regression (Vapnik et al., 1997; Vapnik, 1998), where in place of the classi cation margin, the -insensitive region is introduced, a `tube' of around the function within which errors are not penalised....
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...We consider functions of a type corresponding to those implemented by another sparse linearly-parameterised model, the support vector machine (SVM) (Boser et al., 1992; Vapnik, 1998; Sch olkopf et al., 1999a)....
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Book•
01 Jan 1995TL;DR: This is the first comprehensive treatment of feed-forward neural networks from the perspective of statistical pattern recognition, and is designed as a text, with over 100 exercises, to benefit anyone involved in the fields of neural computation and pattern recognition.
Abstract: From the Publisher:
This is the first comprehensive treatment of feed-forward neural networks from the perspective of statistical pattern recognition. After introducing the basic concepts, the book examines techniques for modelling probability density functions and the properties and merits of the multi-layer perceptron and radial basis function network models. Also covered are various forms of error functions, principal algorithms for error function minimalization, learning and generalization in neural networks, and Bayesian techniques and their applications. Designed as a text, with over 100 exercises, this fully up-to-date work will benefit anyone involved in the fields of neural computation and pattern recognition.
19,056 citations
14,948 citations
Book•
01 Jan 1973
TL;DR: In this article, a unified, comprehensive and up-to-date treatment of both statistical and descriptive methods for pattern recognition is provided, including Bayesian decision theory, supervised and unsupervised learning, nonparametric techniques, discriminant analysis, clustering, preprosessing of pictorial data, spatial filtering, shape description techniques, perspective transformations, projective invariants, linguistic procedures, and artificial intelligence techniques for scene analysis.
Abstract: Provides a unified, comprehensive and up-to-date treatment of both statistical and descriptive methods for pattern recognition. The topics treated include Bayesian decision theory, supervised and unsupervised learning, nonparametric techniques, discriminant analysis, clustering, preprosessing of pictorial data, spatial filtering, shape description techniques, perspective transformations, projective invariants, linguistic procedures, and artificial intelligence techniques for scene analysis.
13,647 citations