| Mu ller, K.-R., Mika, S., Ratsch, G., Tsuda, K., Scholkopf, B., 2001. An introduction to kernel-based learning algorithms. IEEE Trans. Neural Net. 12 (2), 181 -- 201. |
....experts, expectation maximization, support vector machines, unsupervised time series segmentation. I. INTRODUCTION Recently support vector machines (SVMs) 26] have been a promising method for data classification and regression. For an account of various applications of the method, see [15], 16] and references therein. However, its application to unsupervised learning problems has not been exploited much. In this paper we aim to apply it to unsupervised segmentation of time series. Practical applications of unsupervised segmentation of time series include, for example, speech ....
K.-R. M uller, S. Mika, G. R atsch, K. Tsuda, and B. Sch olkopf. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 12(2):181--201, 2001.
....provide good performance when the data distribution is close to Gaussian. Other classification algorithms determine their decision boundaries by supporting samples [6] Supporting samples are typically training samples near the decision boundary. Support vector machines (SVM) 7] 8] 9] 10][11] optimize the supporting samples by maximizing the gap between the authentic samples and the imposter samples regardless of the data distribution. This maxrain approach outperforms the global optimization approach, when the data distribution is not Gaussian as shown by several face ....
K.-R. Mtiller, S. Mika, G. Riitsch,, K. Tsuda, B. Sch61kopf, "An Introduction to kernel-based learning algorithms", IEEE Transactions on Neural Networks, Vol. 12, pp.181 201, 2001.
....provide good performance when the data distribution is close to Gaussian. Other matching algorithms determine their decision boundaries by supporting patterns [10] Supporting patterns are typically training samples near the decision boundary. Support vector machines (SVM) 11] 12] 13] 14][15] optimize the supporting patterns by maximizing the gap between the authentic samples and the imposter samples regardless of the data distribution. This maxmin approach outperforms the global optimization approach, when the data distribution is not Gaussian as shown by several face ....
K.-R. Miller, S. Mika, G. Ritsch,, K. Tsuda, B. Sch61kopf, "An Introduction to kernel-based learning algorithms", IEEE Transactions on Neural Networks, Vol. 12, 2001, pp.181 201.
....the system. Our work in the following months will be to propose solutions to these problems, including: Finding more appropriate methods for dimensionality reduction and parameter selection.We might have to modify the PCA method or apply other nonlinear transformations (for example Kernel PCA [21] or Locally Linear Embedding [26] These dimensionality reduction methods might be applied not only to the input data space but also to the Fisher score space. Changing the parameters of the training algorithm of SVM so that the optimization criterion becomes HTER. The GMM model (can be ....
K. R. Muller, S. Mika, G. Ratsch, K. Tsuda, and B. Scholkopf. An introduction to kernel-based learning algorithms. IEEETNN: IEEE Transactions on Neural Networks, 12:181201, 2001.
....available. 1 Some strategies for approximating SVM posterior estimates in a post processing step have been reported in the literature, see e.g. 3, 4] In this paper, however, we restrict our attention to fully probabilistic models. 2 A recent overview over kernel methods can be found in [8]. 2 Kernelized Logistic Regression The problem of classification formally consists of assigning observed vectors x # R d into one of c classes. A classifier is a mapping that assigns labels to observations. In practice, a classifier is trained on a set of observed i.i.d. data label pairs ....
K.-R. M uller, S. Mika, G. R atsch, K. Tsuda, and B. Sch olkopf. An introduction to kernelbased learning algorithms. IEEE Transactions on Neural Networks, 12(2):181--201, March 2001.
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K.-R. Mller, S. Mika, G. Rtsch, K. Tsuda, and B. Schlkopf, "An introduction to kernel-based learning algorithms," IEEE Transactions on Neural Networks, vol. 12, no. 2, pp. 181--201, 2001.
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Mller, K.-R., Mika, S., Rtsch, G., Tsuda, K., and Schlkopf, B. (2001). An introduction to kernel-based learning algorithms. IEEE Neural Networks, 12(2):181--201.
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K.-R. Muller, S. Mika, G. Ratsch, K. Tsuda, and B. Scholkopf. An introduction to kernelbased learning algorithms. IEEE Transactions on Neural Networks, 12(2):181--201, 2001.
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K.-R. Mller, S. Mika, G. Rtsch, K. Tsuda, and B. Schlkopf, "An introduction to kernel-based learning algorithms, " IEEE Neural Networks, vol. 12, no. 2, pp. 181-- 201, May 2001.
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K.-R. Muller, S. Mika, G. Ratsch, K. Tsuda, and B. Scholkopf. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 2001. in Press.
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K.-R. Mller, S. Mika, G. Rtsch, K. Tsuda, and B. Schlkopf, "An introduction to kernel-based learning algorithms," IEEE Neural Networks, vol. 12, no. 2, pp. 181--201, May 2001. [Online]. Available: http://www.first.gmd.de/persons/Mueller.Klaus-Robert/review.ps.gz
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K.-R. Muller, S. Mika, G. Ratsch, K. Tsuda, and B. Scholkopf. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 12(2):181--201, 2001.
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K.-R. Muller, S. Mika, G. Ratsch, K. Tsuda, and B. Scholkopf. An introduction to kernel-based learning algorithms. IEEE Trans. Neural Networks, 12(2):181--201, 2001.
....that we will address in this paper, has an even more challenging setup. Here, the mixing model reads x[t] f (s[t] 2) and f is an (at least approximately invertible) nonlinear function from to . First algorithms for this problem are based on the idea of kernel based learning (cf. e.g. [22, 6, 16]) were only tried on toy signals [8] The difference between our kTDSEP algorithm and [8] lies mainly in the manner and the superior efficiency in which the kernel feature space is constructed and used for unmixing (our approach considers temporal decorrelation) This eventually allows to demix ....
....data sets that are nonlinearly mixed according to Eq. 2) Let us first introduce the basic ideas of kernel based methods that are needed for our algorithm. For input vectors x[t] t = 1 . T ) from an input space a kernel function k : # that fulfills certain conditions (cf. [16]) induces a mapping # : into some feature such that the dot product for points in the image of # can be simply calculated using the kernel function (often called the kernel trick) k(x[i] x[j] #(x[i] #(x[j] 3) By using linear algorithms in feature space, nonlinear problems in ....
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K.-R. Mller, S. Mika, G. Rtsch, K. Tsuda, and B. Schlkopf. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 12(2):181--201, 2001.
....models [5, 15, 14] the number of parameters is much larger than the number of classes. However, in supervised scenarios, the existence of nuisance dimensions is not a serious problem, because advanced supervised classifiers such as the support vector machine have a built in feature selector [7]. However in unsupervised scenarios without class labels, it is much more difficult to ignore nuisance dimensions. Fig. 2 shows how the feature space looks like, when the number of clusters is two and only one nuisance dimension is involved. Projected on the important dimension, clusters will be ....
....Concluding Remarks In this paper, we illustrated how the class information is encoded in the Fisher score: most information is packed in a few dimensions and there are a lot of nuisance dimensions. Advanced supervised classifiers such as the support vector machine have a built in feature selector [7], so they can detect important dimensions automatically. However in unsupervised learning, it is not easy to detect important dimensions because of the lack of class labels. We proposed a novel very simple clustering algorithm that can ignore nuisance dimensions and tested it in artificial and ....
K.-R. Muller, S. Mika, G. Ratsch, K. Tsuda, and B. Scholkopf. An introduction to kernel-based learning algorithms. IEEE Trans. Neural Networks, 12(2):181--201, 2001.
....some remarks. 2 Reviewing SVMs, LPMs and Selected LOO Bounds When learning with SVMs [7] and LPMs [1] one is seeking for the coefficients of a linear combination of kernel functions K(xi, i.e. f,b(x) b i=1 iK(zci,zc) This is done by solving the following type of optimization problem [4]: C Ei:li with yif,b(xi) l i, i:l, g, 1) where II lip is the 2 norm of c in feature space for SVMs and the I norm of c (in the coefficient space) for LPMs, respectively. The data and labels are denoted by iI i n,y 1, 1 respectively. C is the regularization parameter. When solving (1) ....
K.-R. Mfiller, S. Mika, G. Rtsch, K. Tsuda, and B. SchSlkopf. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 2001. in Press.
....motor cortices which increase prior to the keystroke, cf. Figure 2. But here the selection is automatically adapted to subject, electrode placement, etc. Our implementation of RFD and SFD uses the cplex optimizer [15] Support Vector Machines (SVMs) are well known for their use with kernels [16, 17]. Here we only consider linear SVMs: y k (w # x k b) 1 x k , and x k 0 The choice of regulization keeps a bound on the Vapnik Chervonenkis dimension small. In an equivalent formulation the objective is to maximize the margin between the two classes (while minimizing the soft ....
.... our future research activities will therefore focus on (a) projection techniques like ICA, b) time series approaches to capture the (non linear) dynamics of the multivariate EEG signals, and (c) construction of specially adapted kernel functions (SVM or kernel FD) in the spirit of, e.g. [17] to ultimately obtain a BCI feedback system with an even higher bit rate and accuracy. Acknowledgements. We thank S. Harmeling, M. Kawanabe, J. Kohlmorgen, J. Laub, S. Mika, G. Rtsch, R. Vigrio and A. Ziehe for helpful discussions. ....
K.-R. Mller, S. Mika, G. Rtsch, K. Tsuda, and B. Schlkopf, "An Introduction to KernelBased Learning Algorithms", IEEE Transactions on Neural Networks, 12(2): 181--201, 2001.
....of our kTDSEP algorithm for the problem of nonlinear BSS. 1 Introduction In a widespread area of applications kernel based learning machines, e.g. Support Vector Machines (e.g. 19, 6] give excellent solutions. This holds both for problems of supervised and unsupervised learning (e.g. [3, 16, 12]) The general idea is to map the data x i (i = 1, T ) into some kernel feature space by some mapping # : # F . Performing a simple linear algorithm in , then corresponds to a nonlinear algorithm in input space. Essential ingredients to kernel based learning are (a) VC theory ....
K.-R. Mller, S. Mika, G. Rtsch, K. Tsuda, and B. Schlkopf. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 12(2):181--201, 2001.
.... [7] 8] 9] Their high accuracy, ease of implementation, and wide applicability has placed them in the standard toolbox of machine learning, next to neural networks [10] 11] 12] and kernel based learning methods like support vector machines (SVMs) 13] 14] 15] 16] 17] 18] 19] [20]. The present paper focuses on two points. On the algorithmic side we will propose a boosting like one class classification algorithm based on a technique called barrier optimization (standard books on optimization, e.g. 21] In one class classification one trains on unlabeled data, trying to ....
....j th dimension in feature space. 4 III. FROM ONE CLASS SVMS TO ONE CLASS LPS A. One Class SVMs Over the last years, SVMs have been generalized in various ways to deal with a number of different learning problems, e.g. also the problem of unsupervised learning, i.e. where the data are unlabeled [20], 41] One could hold the view that un supervised learning is essentially synonymous with density estimation, for, once a density has been estimated, all other statistical properties of the regularity underlying the data can readily be deduced. However, density estimation in high dimensional ....
K.-R. Miller, S. Mika, G. Rtsch, K. Tsuda, and B. Sch61kopf, "An introduction to kernel-based learning algorithms," IEEE Transactions on Neural Networks, vol. 12, no. 2, pp. 181 201, 2001.
....some remarks. 2 Reviewing SVMs, LPMs and Selected LOO Bounds When learning with SVMs [7] and LPMs [1] one is seeking for the coefficients of a linear combination of kernel functions K(xi, i.e. f,b(x) b i=1 (iK(lri,lr) This is done by solving the following type of optimization problem [4]: min IlsliP q Ei=li with Yifot,b(11i) 1 i, i = 1, g, 1) b, o where ] lip is the 2 norm of e in feature space for SVMs and the I norm of e (in the coefficient space) for LPMs, respectively. The data and labels are denoted by iI i n,y 1, 1 respectively. C is the regularization ....
K.-R. Mfiller, S. Mika, G. RStsch, K. Tsuda, and B. SchSlkopf. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 2001. in Press.
....[2] and are referred to as pairwise coupling. Learning such pairwise decision rules may be a much simpler problem than separating each class from the others. In this paper special emphasis is put on nonlinear Mercer kernel based classifiers. A recent overview over kernel methods can be found in [3]. Since for kernel methods the computational efficiency is mostly determined by the number of training samples, the pairwise coupling scheme also overcomes the numerical problems of the one against all strategy: it is much easier to solve small problems that to solve large problems. For ....
K.-R. Muller, S. Mika, G. Ratsch, K. Tsuda, and B. Scholkopf, "An introduction to kernel-based learning algorithms," IEEE Transactions on Neural Networks, vol. 12, no. 2, pp. 181--201, March 2001.
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Mu ller, K.-R., Mika, S., Ratsch, G., Tsuda, K., Scholkopf, B., 2001. An introduction to kernel-based learning algorithms. IEEE Trans. Neural Net. 12 (2), 181 -- 201.
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K.-R. M uller, S. Mika, G. R atsch, , and K. Tsuda. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 12(2):181--201, 2001.
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K. R. Muller, S. Mika, G. Ratsch, , and K. Tsuda. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 12(2):181--201, 2001.
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K.-R. M. Mller; G. Ratsch, K. Tsuda, K.; B. Schlkopf, "An Introduction to Kernel-based Learning Algorithms," IEEE Transactions on Neural Networks, 2:181-201, 2001.
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Muller, K. R., Mika, S., Ratsch, G., Tsuda, K., & Scholkopf, B. (2001). An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 12, 181 -- 201.
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K. R. Muller, S. Mika, G. Ratsch, K. Tsuda, B. Scholkopf, "An Introduction to Kernel-based Learning Algorithms", IEEE Transactions on Neural Networks, Volume 12, Issue 2, pp. 181--201, 2001.
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Klaus-Robert Muller and Sebastian Mika and Gunnar Ratsch and Koji Tsuda and Bernhard Scholkopf. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 12(2):181--202, March 2001.
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K.-R. Mu ller, S. Mika, G. Ratsch, K. Tsuda, B. Scholkopf, An introduction to kernel-based learning algorithms, IEEE Transactions on Neural Networks 12 (2) (2001) 181 -- 201.
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K. Muller, S.Mika, G. Ratsch, K. Tsuda, and B. Scholkopf. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 12(2):181--201, March 2001.
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Muller, K., S.Mika, G. Ratsch, K. Tsuda, and B. Scholkopf: 2001, `An introduction to kernel-based learning algorithms'. IEEE Transactions on Neural Networks 12(2), 181--201.
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K. Muller, S.Mika, G. Ratsch, K. Tsuda, and B. Scholkopf, "An introduction to kernel-based learning algorithms," IEEE Transactions on Neural Networks, vol. 12, no. 2, pp. 181--201, March 2001.
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K.R. Mller, S. Mika, G. Rtsch, K. Tsuda, B. Schlkopf, "An Introduction to Kernel-based Learning Algorithms", IEEE Transactions on Neural Networks, 12/2, 181-202, 2001.
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K. R. Muller, S. Mika, G. Ratsch, K. Tsuda, B. Scholkopf, \An Introduction to kernel-based learning algorithms", IEEE Transactions on Neural Networks, Vol. 12, pp. 181-201, 2001.
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K.-R. Muller, S. Mika, G. Ratsch, K. Tsuda, and B. Scholkopf, "An introduction to kernel-based learning algorithms, " IEEE Trans. Neural Netw., vol. 12, no. 2, pp. 181--201, 2001.
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K. Muller, S. Mika, G. Ratsch, K. Tsuda and B. Scholkopf, \An Introduction to Kernel-Based Learning Algorithms," IEEE Transactions on Neural Networks, vol. 12, no. 2, pp. 181-201, March 2001.
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K.R. Muller, S. Mika, G. Ratsch, K. Tsuda, and B. Scholkopf, \An introduction to kernel-based learning algorithms," IEEE Transactions on Neural Networks, vol. 12, no. 2, pp. 181-201, March 2001.
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K.-R. Muller, S. Mika, G. Ratsch, K. Tsuda, and B. Scholkopf, "An Introduction to Kernel-Based Learning Algorithms," IEEE Transactions on Neural Networks, vol. 12, no. 2, pp. 181-202, 2001.
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K.-R. Muller, S. Mika, G. Ratsch, K. Tsuda, and B. Scholkopf, \An Introduction to Kernel-Based Learning Algorithms," IEEE Transactions on Neural Networks, vol. 12, no. 2, pp. 181-202, 2001.
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K.-R. Mller, S. Mika, G. Rtsch, K. Tsuda, and B. Schlkopf. An introduction to kernelbased learning algorithms. IEEE Neural Networks, 12(2):181201, May 2001.
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K.-R. Muller, S. Mika, G. Ratsch, K. Tsuda, and Scholkopf. An introduction to kernel-based learning algorithms. IEEE Transaction on Neural Networks, 12(2):181--202, March 2001.
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K.-R. M uller, S. Mika, G. R atsch, K. Tsuda, and B. Sch olkopf, "An introduction to kernel-based learning algorithms," IEEE Trans. Neural Network, vol. 12, pp. 181-201, 2001.
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K.R.Muller, S. Mika, G. Ratsch, K. Tsuda, and B. Scholkopf, "An introduction to kernelbased learning algorithm2 IEEE Trans. Neural Networks, vol. 12, no. 2, pp. 181 - 201, Mar. 2001.
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K.-R. Muller, S. Mika, G. Ratsch,, K. Tsuda, B. Scholkopf, \An Introduction to kernel-based learning algorithms", IEEE Transactions on Neural Networks, Vol. 12, pp. 181-201, 2001.
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K. Muller, S.Mika, G. Ratsch, K. Tsuda, and B. Scholkopf, "An introduction to kernel-based learning algorithms," IEEE Transactions on Neural Networks, vol. 12, no. 2, pp. 181--201, March 2001.
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K.R. Mller, S. Mika, G. Rtsch, K. Tsuda, B. Schlkopf, "An Introduction to Kernel-based Learning Algorithms", IEEE Transactions on Neural Networks, 12/2, 181-202, 2001.
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K.-R. Mller, S. Mika, G. Rtsch, K. Tsuda, and B. Schlkopf, "An introduction to kernel-based learning algorithms," IEEE Trans. Neural Networks, vol. 12, pp. 181--202, Apr. 2001.
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K.-R. Mller, S. Mika, G. Rtsch, K. Tsuda, and B. Schlkopf, "An introduction to kernel-based learning algorithms," IEEE Trans. Neural Networks, vol. 12, pp. 181--201, Jan. 2001.
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K.-R. Mller, S. Mika, G. Rtsch, K. Tsuda, and B. Schlkopf. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 12(2):181201, 2001. 20
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K.-R. Muller, S. Mika, G. Ratsch, K. Tsuda, and B. Scholkopf. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 12(2):181--202, 2001.
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