Results 1  10
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213
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 2812 (126 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
Consistency of the group lasso and multiple kernel learning
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2007
"... We consider the leastsquare regression problem with regularization by a block 1norm, i.e., a sum of Euclidean norms over spaces of dimensions larger than one. This problem, referred to as the group Lasso, extends the usual regularization by the 1norm where all spaces have dimension one, where it ..."
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Cited by 283 (35 self)
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We consider the leastsquare regression problem with regularization by a block 1norm, i.e., a sum of Euclidean norms over spaces of dimensions larger than one. This problem, referred to as the group Lasso, extends the usual regularization by the 1norm where all spaces have dimension one, where it is commonly referred to as the Lasso. In this paper, we study the asymptotic model consistency of the group Lasso. We derive necessary and sufficient conditions for the consistency of group Lasso under practical assumptions, such as model misspecification. When the linear predictors and Euclidean norms are replaced by functions and reproducing kernel Hilbert norms, the problem is usually referred to as multiple kernel learning and is commonly used for learning from heterogeneous data sources and for non linear variable selection. Using tools from functional analysis, and in particular covariance operators, we extend the consistency results to this infinite dimensional case and also propose an adaptive scheme to obtain a consistent model estimate, even when the necessary condition required for the non adaptive scheme is not satisfied.
Multicategory Support Vector Machines, theory, and application to the classification of microarray data and satellite radiance data
 Journal of the American Statistical Association
, 2004
"... Twocategory support vector machines (SVM) have been very popular in the machine learning community for classi � cation problems. Solving multicategory problems by a series of binary classi � ers is quite common in the SVM paradigm; however, this approach may fail under various circumstances. We pro ..."
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Cited by 261 (25 self)
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Twocategory support vector machines (SVM) have been very popular in the machine learning community for classi � cation problems. Solving multicategory problems by a series of binary classi � ers is quite common in the SVM paradigm; however, this approach may fail under various circumstances. We propose the multicategory support vector machine (MSVM), which extends the binary SVM to the multicategory case and has good theoretical properties. The proposed method provides a unifying framework when there are either equal or unequal misclassi � cation costs. As a tuning criterion for the MSVM, an approximate leaveoneout crossvalidation function, called Generalized Approximate Cross Validation, is derived, analogous to the binary case. The effectiveness of the MSVM is demonstrated through the applications to cancer classi � cation using microarray data and cloud classi � cation with satellite radiance pro � les.
Correcting sample selection bias by unlabeled data
"... We consider the scenario where training and test data are drawn from different distributions, commonly referred to as sample selection bias. Most algorithms for this setting try to first recover sampling distributions and then make appropriate corrections based on the distribution estimate. We prese ..."
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Cited by 206 (12 self)
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We consider the scenario where training and test data are drawn from different distributions, commonly referred to as sample selection bias. Most algorithms for this setting try to first recover sampling distributions and then make appropriate corrections based on the distribution estimate. We present a nonparametric method which directly produces resampling weights without distribution estimation. Our method works by matching distributions between training and testing sets in feature space. Experimental results demonstrate that our method works well in practice.
Measuring statistical dependence with HilbertSchmidt norms
 PROCEEDINGS ALGORITHMIC LEARNING THEORY
, 2005
"... We propose an independence criterion based on the eigenspectrum of covariance operators in reproducing kernel Hilbert spaces (RKHSs), consisting of an empirical estimate of the HilbertSchmidt norm of the crosscovariance operator (we term this a HilbertSchmidt Independence Criterion, or HSIC). Th ..."
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Cited by 158 (44 self)
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We propose an independence criterion based on the eigenspectrum of covariance operators in reproducing kernel Hilbert spaces (RKHSs), consisting of an empirical estimate of the HilbertSchmidt norm of the crosscovariance operator (we term this a HilbertSchmidt Independence Criterion, or HSIC). This approach has several advantages, compared with previous kernelbased independence criteria. First, the empirical estimate is simpler than any other kernel dependence test, and requires no userdefined regularisation. Second, there is a clearly defined population quantity which the empirical estimate approaches in the large sample limit, with exponential convergence guaranteed between the two: this ensures that independence tests based on HSIC do not suffer from slow learning rates. Finally, we show in the context of independent component analysis (ICA) that the performance of HSIC is competitive with that of previously published kernelbased criteria, and of other recently published ICA methods.
A kernel method for the twosampleproblem
 In NIPS. 2006
"... We propose two statistical tests to determine if two samples are from different distributions. Our test statistic is in both cases the distance between the means of the two samples mapped into a reproducing kernel Hilbert space (RKHS). The first test is based on a large deviation bound for the test ..."
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Cited by 118 (30 self)
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We propose two statistical tests to determine if two samples are from different distributions. Our test statistic is in both cases the distance between the means of the two samples mapped into a reproducing kernel Hilbert space (RKHS). The first test is based on a large deviation bound for the test statistic, while the second is based on the asymptotic distribution of this statistic. The test statistic can be computed in O(m2) time. We apply our approach to a variety of problems, including attribute matching for databases using the Hungarian marriage method, where our test performs strongly. We also demonstrate excellent performance when comparing distributions over graphs, for which no alternative tests currently exist. 1
A Hilbert space embedding for distributions
 In Algorithmic Learning Theory: 18th International Conference
, 2007
"... Abstract. We describe a technique for comparing distributions without the need for density estimation as an intermediate step. Our approach relies on mapping the distributions into a reproducing kernel Hilbert space. Applications of this technique can be found in twosample tests, which are used for ..."
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Cited by 110 (45 self)
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Abstract. We describe a technique for comparing distributions without the need for density estimation as an intermediate step. Our approach relies on mapping the distributions into a reproducing kernel Hilbert space. Applications of this technique can be found in twosample tests, which are used for determining whether two sets of observations arise from the same distribution, covariate shift correction, local learning, measures of independence, and density estimation. Kernel methods are widely used in supervised learning [1, 2, 3, 4], however they are much less established in the areas of testing, estimation, and analysis of probability distributions, where information theoretic approaches [5, 6] have long been dominant. Recent examples include [7] in the context of construction of graphical models, [8] in the context of feature extraction, and [9] in the context of independent component analysis. These methods have by and large a common issue: to compute quantities such as the mutual information, entropy, or KullbackLeibler divergence, we require sophisticated space partitioning and/or
Domain Adaptation via Transfer Component Analysis
"... Domain adaptation solves a learning problem in a target domain by utilizing the training data in a different but related source domain. Intuitively, discovering a good feature representation across domains is crucial. In this paper, we propose to find such a representation through a new learning met ..."
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Cited by 102 (18 self)
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Domain adaptation solves a learning problem in a target domain by utilizing the training data in a different but related source domain. Intuitively, discovering a good feature representation across domains is crucial. In this paper, we propose to find such a representation through a new learning method, transfer component analysis (TCA), for domain adaptation. TCA tries to learn some transfer components across domains in a Reproducing Kernel Hilbert Space (RKHS) using Maximum Mean Discrepancy (MMD). In the subspace spanned by these transfer components, data distributions in different domains are close to each other. As a result, with the new representations in this subspace, we can apply standard machine learning methods to train classifiers or regression models in the source domain for use in the target domain. The main contribution of our work is that we propose a novel feature representation in which to perform domain adaptation via a new parametric kernel using feature extraction methods, which can dramatically minimize the distance between domain distributions by projecting data onto the learned transfer components. Furthermore, our approach can handle large datsets and naturally lead to outofsample generalization. The effectiveness and efficiency of our approach in are verified by experiments on two realworld applications: crossdomain indoor WiFi localization and crossdomain text classification. 1
A kernel statistical test of independence
, 2008
"... Although kernel measures of independence have been widely applied in machine learning (notably in kernel ICA), there is as yet no method to determine whether they have detected statistically significant dependence. We provide a novel test of the independence hypothesis for one particular kernel inde ..."
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Cited by 95 (48 self)
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Although kernel measures of independence have been widely applied in machine learning (notably in kernel ICA), there is as yet no method to determine whether they have detected statistically significant dependence. We provide a novel test of the independence hypothesis for one particular kernel independence measure, the HilbertSchmidt independence criterion (HSIC). The resulting test costs O(m 2), where m is the sample size. We demonstrate that this test outperforms established contingency table and functional correlationbased tests, and that this advantage is greater for multivariate data. Finally, we show the HSIC test also applies to text (and to structured data more generally), for which no other independence test presently exists. 1
A Review of Kernel Methods in Machine Learning
, 2006
"... We review recent methods for learning with positive definite kernels. All these methods formulate learning and estimation problems as linear tasks in a reproducing kernel Hilbert space (RKHS) associated with a kernel. We cover a wide range of methods, ranging from simple classifiers to sophisticate ..."
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Cited by 95 (4 self)
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We review recent methods for learning with positive definite kernels. All these methods formulate learning and estimation problems as linear tasks in a reproducing kernel Hilbert space (RKHS) associated with a kernel. We cover a wide range of methods, ranging from simple classifiers to sophisticated methods for estimation with structured data.