Results 1  10
of
161
Online Learning with Kernels
, 2003
"... Kernel based algorithms such as support vector machines have achieved considerable success in various problems in the batch setting where all of the training data is available in advance. Support vector machines combine the socalled kernel trick with the large margin idea. There has been little u ..."
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Cited by 2029 (128 self)
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Kernel based algorithms such as support vector machines have achieved considerable success in various problems in the batch setting where all of the training data is available in advance. Support vector machines combine the socalled kernel trick with the large margin idea. There has been little use of these methods in an online setting suitable for realtime applications. In this paper we consider online learning in a Reproducing Kernel Hilbert Space. By considering classical stochastic gradient descent within a feature space, and the use of some straightforward tricks, we develop simple and computationally efficient algorithms for a wide range of problems such as classification, regression, and novelty detection. In addition to allowing the exploitation of the kernel trick in an online setting, we examine the value of large margins for classification in the online setting with a drifting target. We derive worst case loss bounds and moreover we show the convergence of the hypothesis to the minimiser of the regularised risk functional. We present some experimental results that support the theory as well as illustrating the power of the new algorithms for online novelty detection. In addition
Sparse Bayesian Learning and the Relevance Vector Machine
, 2001
"... This paper introduces a general Bayesian framework for obtaining sparse solutions to regression and classication 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 vec ..."
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Cited by 552 (5 self)
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This paper introduces a general Bayesian framework for obtaining sparse solutions to regression and classication 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 stateoftheart `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 oering a number of additional advantages. These include the benets of probabilistic predictions, automatic estimation of `nuisance' parameters, and the facility to utilise arbitrary basis functions (e.g. non`Mercer' kernels).
Gaussian processes for machine learning
 in: Adaptive Computation and Machine Learning
, 2006
"... Abstract. We give a basic introduction to Gaussian Process regression models. We focus on understanding the role of the stochastic process and how it is used to define a distribution over functions. We present the simple equations for incorporating training data and examine how to learn the hyperpar ..."
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Cited by 280 (2 self)
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Abstract. We give a basic introduction to Gaussian Process regression models. We focus on understanding the role of the stochastic process and how it is used to define a distribution over functions. We present the simple equations for incorporating training data and examine how to learn the hyperparameters using the marginal likelihood. We explain the practical advantages of Gaussian Process and end with conclusions and a look at the current trends in GP work. Supervised learning in the form of regression (for continuous outputs) and classification (for discrete outputs) is an important constituent of statistics and machine learning, either for analysis of data sets, or as a subgoal of a more complex problem. Traditionally parametric 1 models have been used for this purpose. These have a possible advantage in ease of interpretability, but for complex data sets, simple parametric models may lack expressive power, and their more complex counterparts (such as feed forward neural networks) may not be easy to work with
Kernels and Regularization on Graphs
, 2003
"... We introduce a family of kernels on graphs based on the notion of regularization operators. This generalizes in a natural way the notion of regularization and Greens functions, as commonly used for real valued functions, to graphs. It turns out that di#usion kernels can be found as a special cas ..."
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Cited by 168 (9 self)
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We introduce a family of kernels on graphs based on the notion of regularization operators. This generalizes in a natural way the notion of regularization and Greens functions, as commonly used for real valued functions, to graphs. It turns out that di#usion kernels can be found as a special case of our reasoning. We show that the class of positive, monotonically decreasing functions on the unit interval leads to kernels and corresponding regularization operators.
Learning with Labeled and Unlabeled Data
, 2001
"... In this paper, on the one hand, we aim to give a review on literature dealing with the problem of supervised learning aided by additional unlabeled data. On the other hand, being a part of the author's first year PhD report, the paper serves as a frame to bundle related work by the author as well as ..."
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Cited by 165 (3 self)
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In this paper, on the one hand, we aim to give a review on literature dealing with the problem of supervised learning aided by additional unlabeled data. On the other hand, being a part of the author's first year PhD report, the paper serves as a frame to bundle related work by the author as well as numerous suggestions for potential future work. Therefore, this work contains more speculative and partly subjective material than the reader might expect from a literature review. We give a rigorous definition of the problem and relate it to supervised and unsupervised learning. The crucial role of prior knowledge is put forward, and we discuss the important notion of inputdependent regularization. We postulate a number of baseline methods, being algorithms or algorithmic schemes which can more or less straightforwardly be applied to the problem, without the need for genuinely new concepts. However, some of them might serve as basis for a genuine method. In the literature revi...
A Generalized Representer Theorem
 In Proceedings of the Annual Conference on Computational Learning Theory
, 2001
"... Wahba's classical representer theorem states that the solutions of certain risk minimization problems involving an empirical risk term and a quadratic regularizer can be written as expansions in terms of the training examples. We generalize the theorem to a larger class of regularizers and empir ..."
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Cited by 136 (17 self)
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Wahba's classical representer theorem states that the solutions of certain risk minimization problems involving an empirical risk term and a quadratic regularizer can be written as expansions in terms of the training examples. We generalize the theorem to a larger class of regularizers and empirical risk terms, and give a selfcontained proof utilizing the feature space associated with a kernel. The result shows that a wide range of problems have optimal solutions that live in the finite dimensional span of the training examples mapped into feature space, thus enabling us to carry out kernel algorithms independent of the (potentially infinite) dimensionality of the feature space.
Probabilistic nonlinear principal component analysis with Gaussian process latent variable models
 Journal of Machine Learning Research
, 2005
"... Summarising a high dimensional data set with a low dimensional embedding is a standard approach for exploring its structure. In this paper we provide an overview of some existing techniques for discovering such embeddings. We then introduce a novel probabilistic interpretation of principal component ..."
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Cited by 134 (14 self)
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Summarising a high dimensional data set with a low dimensional embedding is a standard approach for exploring its structure. In this paper we provide an overview of some existing techniques for discovering such embeddings. We then introduce a novel probabilistic interpretation of principal component analysis (PCA) that we term dual probabilistic PCA (DPPCA). The DPPCA model has the additional advantage that the linear mappings from the embedded space can easily be nonlinearised through Gaussian processes. We refer to this model as a Gaussian process latent variable model (GPLVM). Through analysis of the GPLVM objective function, we relate the model to popular spectral techniques such as kernel PCA and multidimensional scaling. We then review a practical algorithm for GPLVMs in the context of large data sets and develop it to also handle discrete valued data and missing attributes. We demonstrate the model on a range of realworld and artificially generated data sets.
Kernel partial least squares regression in reproducing kernel hilbert space
 Journal of Machine Learning Research
, 2001
"... A family of regularized least squares regression models in a Reproducing Kernel Hilbert Space is extended by the kernel partial least squares (PLS) regression model. Similar to principal components regression (PCR), PLS is a method based on the projection of input (explanatory) variables to the late ..."
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Cited by 106 (5 self)
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A family of regularized least squares regression models in a Reproducing Kernel Hilbert Space is extended by the kernel partial least squares (PLS) regression model. Similar to principal components regression (PCR), PLS is a method based on the projection of input (explanatory) variables to the latent variables (components). However, in contrast to PCR, PLS creates the components by modeling the relationship between input and output variables while maintaining most of the information in the input variables. PLS is useful in situations where the number of explanatory variables exceeds the number of observations and/or a high level of multicollinearity among those variables is assumed. Motivated by this fact we will provide a kernel PLS algorithm for construction of nonlinear regression models in possibly highdimensional feature spaces. We give the theoretical description of the kernel PLS algorithm and we experimentally compare the algorithm with the existing kernel PCR and kernel ridge regression techniques. We will demonstrate that on the data sets employed kernel PLS achieves the same results as kernel PCR but uses significantly fewer, qualitatively different components. 1.
Adaptive Sparseness for Supervised Learning
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2003
"... The goal of supervised learning is to infer a functional mapping based on a set of training examples. To achieve good generalization, it is necessary to control the "complexity" of the learned function. In Bayesian approaches, this is done by adopting a prior for the parameters of the function bei ..."
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Cited by 80 (4 self)
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The goal of supervised learning is to infer a functional mapping based on a set of training examples. To achieve good generalization, it is necessary to control the "complexity" of the learned function. In Bayesian approaches, this is done by adopting a prior for the parameters of the function being learned. We propose a Bayesian approach to supervised learning, which leads to sparse solutions; that is, in which irrelevant parameters are automatically set exactly to zero. Other ways to obtain sparse classifiers (such as Laplacian priors, support vector machines) involve (hyper)parameters which control the degree of sparseness of the resulting classifiers; these parameters have to be somehow adjusted/estimated from the training data. In contrast, our approach does not involve any (hyper)parameters to be adjusted or estimated. This is achieved by a hierarchicalBayes interpretation of the Laplacian prior, which is then modified by the adoption of a Jeffreys' noninformative hyperprior. Implementation is carried out by an expectationmaximization (EM) algorithm. Experiments with several benchmark data sets show that the proposed approach yields stateoftheart performance. In particular, our method outperforms SVMs and performs competitively with the best alternative techniques, although it involves no tuning or adjustment of sparsenesscontrolling hyperparameters.
A Bayesian Committee Machine
 NEURAL COMPUTATION
, 2000
"... The Bayesian committee machine (BCM) is a novel approach to combining estimators which were trained on different data sets. Although the BCM can be applied to the combination of any kind of estimators the main foci are Gaussian process regression and related systems such as regularization networks a ..."
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Cited by 71 (7 self)
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The Bayesian committee machine (BCM) is a novel approach to combining estimators which were trained on different data sets. Although the BCM can be applied to the combination of any kind of estimators the main foci are Gaussian process regression and related systems such as regularization networks and smoothing splines for which the degrees of freedom increase with the number of training data. Somewhat surprisingly, we nd that the performance of the BCM improves if several test points are queried at the same time and is optimal if the number of test points is at least as large as the degrees of freedom of the estimator. The BCM also provides a new solution for online learning with potential applications to data mining. We apply the BCM to systems with fixed basis functions and discuss its relationship to Gaussian process regression. Finally, we also show how the ideas behind the BCM can be applied in a nonBayesian setting to extend the input dependent combination of estimators.