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2012


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High Gamma-Power Predicts Performance in Brain-Computer Interfacing

Grosse-Wentrup, M., Schölkopf, B.

(3), Max-Planck-Institut für Intelligente Systeme, Tübingen, February 2012 (techreport)

Abstract
Subjects operating a brain-computer interface (BCI) based on sensorimotor rhythms exhibit large variations in performance over the course of an experimental session. Here, we show that high-frequency gamma-oscillations, originating in fronto-parietal networks, predict such variations on a trial-to-trial basis. We interpret this nding as empirical support for an in uence of attentional networks on BCI-performance via modulation of the sensorimotor rhythm.

PDF [BibTex]

2012

PDF [BibTex]

2007


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Bayesian Estimators for Robins-Ritov’s Problem

Harmeling, S., Toussaint, M.

(EDI-INF-RR-1189), School of Informatics, University of Edinburgh, October 2007 (techreport)

Abstract
Bayesian or likelihood-based approaches to data analysis became very popular in the field of Machine Learning. However, there exist theoretical results which question the general applicability of such approaches; among those a result by Robins and Ritov which introduce a specific example for which they prove that a likelihood-based estimator will fail (i.e. it does for certain cases not converge to a true parameter estimate, even given infinite data). In this paper we consider various approaches to formulate likelihood-based estimators in this example, basically by considering various extensions of the presumed generative model of the data. We can derive estimators which are very similar to the classical Horvitz-Thompson and which also account for a priori knowledge of an observation probability function.

PDF [BibTex]

2007

PDF [BibTex]


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Learning with Transformation Invariant Kernels

Walder, C., Chapelle, O.

(165), Max Planck Institute for Biological Cybernetics, Tübingen, Germany, September 2007 (techreport)

Abstract
Abstract. This paper considers kernels invariant to translation, rotation and dilation. We show that no non-trivial positive definite (p.d.) kernels exist which are radial and dilation invariant, only conditionally positive definite (c.p.d.) ones. Accordingly, we discuss the c.p.d. case and provide some novel analysis, including an elementary derivation of a c.p.d. representer theorem. On the practical side, we give a support vector machine (s.v.m.) algorithm for arbitrary c.p.d. kernels. For the thin-plate kernel this leads to a classifier with only one parameter (the amount of regularisation), which we demonstrate to be as effective as an s.v.m. with the Gaussian kernel, even though the Gaussian involves a second parameter (the length scale).

PDF [BibTex]

PDF [BibTex]


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Scalable Semidefinite Programming using Convex Perturbations

Kulis, B., Sra, S., Jegelka, S.

(TR-07-47), University of Texas, Austin, TX, USA, September 2007 (techreport)

Abstract
Several important machine learning problems can be modeled and solved via semidefinite programs. Often, researchers invoke off-the-shelf software for the associated optimization, which can be inappropriate for many applications due to computational and storage requirements. In this paper, we introduce the use of convex perturbations for semidefinite programs (SDPs). Using a particular perturbation function, we arrive at an algorithm for SDPs that has several advantages over existing techniques: a) it is simple, requiring only a few lines of MATLAB, b) it is a first-order method which makes it scalable, c) it can easily exploit the structure of a particular SDP to gain efficiency (e.g., when the constraint matrices are low-rank). We demonstrate on several machine learning applications that the proposed algorithm is effective in finding fast approximations to large-scale SDPs.

PDF [BibTex]

PDF [BibTex]


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Sparse Multiscale Gaussian Process Regression

Walder, C., Kim, K., Schölkopf, B.

(162), Max Planck Institute for Biological Cybernetics, Tübingen, Germany, August 2007 (techreport)

Abstract
Most existing sparse Gaussian process (g.p.) models seek computational advantages by basing their computations on a set of m basis functions that are the covariance function of the g.p. with one of its two inputs fixed. We generalise this for the case of Gaussian covariance function, by basing our computations on m Gaussian basis functions with arbitrary diagonal covariance matrices (or length scales). For a fixed number of basis functions and any given criteria, this additional flexibility permits approximations no worse and typically better than was previously possible. Although we focus on g.p. regression, the central idea is applicable to all kernel based algorithms, such as the support vector machine. We perform gradient based optimisation of the marginal likelihood, which costs O(m2n) time where n is the number of data points, and compare the method to various other sparse g.p. methods. Our approach outperforms the other methods, particularly for the case of very few basis functions, i.e. a very high sparsity ratio.

PDF [BibTex]

PDF [BibTex]


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Efficient Subwindow Search for Object Localization

Blaschko, M., Hofmann, T., Lampert, C.

(164), Max Planck Institute for Biological Cybernetics, Tübingen, Germany, August 2007 (techreport)

Abstract
Recent years have seen huge advances in object recognition from images. Recognition rates beyond 95% are the rule rather than the exception on many datasets. However, most state-of-the-art methods can only decide if an object is present or not. They are not able to provide information on the object location or extent within in the image. We report on a simple yet powerful scheme that extends many existing recognition methods to also perform localization of object bounding boxes. This is achieved by maximizing the classification score over all possible subrectangles in the image. Despite the impression that this would be computationally intractable, we show that in many situations efficient algorithms exist which solve a generalized maximum subrectangle problem. We show how our method is applicable to a variety object detection frameworks and demonstrate its performance by applying it to the popular bag of visual words model, achieving competitive results on the PASCAL VOC 2006 dataset.

PDF [BibTex]

PDF [BibTex]


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Cluster Identification in Nearest-Neighbor Graphs

Maier, M., Hein, M., von Luxburg, U.

(163), Max-Planck-Institute for Biological Cybernetics, Tübingen, Germany, May 2007 (techreport)

Abstract
Assume we are given a sample of points from some underlying distribution which contains several distinct clusters. Our goal is to construct a neighborhood graph on the sample points such that clusters are ``identified‘‘: that is, the subgraph induced by points from the same cluster is connected, while subgraphs corresponding to different clusters are not connected to each other. We derive bounds on the probability that cluster identification is successful, and use them to predict ``optimal‘‘ values of k for the mutual and symmetric k-nearest-neighbor graphs. We point out different properties of the mutual and symmetric nearest-neighbor graphs related to the cluster identification problem.

PDF [BibTex]

PDF [BibTex]


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Exploring model selection techniques for nonlinear dimensionality reduction

Harmeling, S.

(EDI-INF-RR-0960), School of Informatics, University of Edinburgh, March 2007 (techreport)

Abstract
Nonlinear dimensionality reduction (NLDR) methods have become useful tools for practitioners who are faced with the analysis of high-dimensional data. Of course, not all NLDR methods are equally applicable to a particular dataset at hand. Thus it would be useful to come up with model selection criteria that help to choose among different NLDR algorithms. This paper explores various approaches to this problem and evaluates them on controlled data sets. Comprehensive experiments will show that model selection scores based on stability are not useful, while scores based on Gaussian processes are helpful for the NLDR problem.

PDF Web [BibTex]

PDF Web [BibTex]


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Dirichlet Mixtures of Bayesian Linear Gaussian State-Space Models: a Variational Approach

Chiappa, S., Barber, D.

(161), Max Planck Institute for Biological Cybernetics, Tübingen, Germany, March 2007 (techreport)

Abstract
We describe two related models to cluster multidimensional time-series under the assumption of an underlying linear Gaussian dynamical process. In the first model, times-series are assigned to the same cluster when they show global similarity in their dynamics, while in the second model times-series are assigned to the same cluster when they show simultaneous similarity. Both models are based on Dirichlet Mixtures of Bayesian Linear Gaussian State-Space Models in order to (semi) automatically determine an appropriate number of components in the mixture, and to additionally bias the components to a parsimonious parameterization. The resulting models are formally intractable and to deal with this we describe a deterministic approximation based on a novel implementation of Variational Bayes.

PDF [BibTex]

PDF [BibTex]


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Modeling data using directional distributions: Part II

Sra, S., Jain, P., Dhillon, I.

(TR-07-05), University of Texas, Austin, TX, USA, February 2007 (techreport)

Abstract
High-dimensional data is central to most data mining applications, and only recently has it been modeled via directional distributions. In [Banerjee et al., 2003] the authors introduced the use of the von Mises-Fisher (vMF) distribution for modeling high-dimensional directional data, particularly for text and gene expression analysis. The vMF distribution is one of the simplest directional distributions. TheWatson, Bingham, and Fisher-Bingham distributions provide distri- butions with an increasing number of parameters and thereby commensurately increased modeling power. This report provides a followup study to the initial development in [Banerjee et al., 2003] by presenting Expectation Maximization (EM) procedures for estimating parameters of a mixture of Watson (moW) distributions. The numerical challenges associated with parameter estimation for both of these distributions are significantly more difficult than for the vMF distribution. We develop new numerical approximations for estimating the parameters permitting us to model real- life data more accurately. Our experimental results establish that for certain data sets improved modeling power translates into better results.

PDF [BibTex]

PDF [BibTex]


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Automatic 3D Face Reconstruction from Single Images or Video

Breuer, P., Kim, K., Kienzle, W., Blanz, V., Schölkopf, B.

(160), Max Planck Institute for Biological Cybernetics, Tübingen, Germany, February 2007 (techreport)

Abstract
This paper presents a fully automated algorithm for reconstructing a textured 3D model of a face from a single photograph or a raw video stream. The algorithm is based on a combination of Support Vector Machines (SVMs) and a Morphable Model of 3D faces. After SVM face detection, individual facial features are detected using a novel regression-and classification-based approach, and probabilistically plausible configurations of features are selected to produce a list of candidates for several facial feature positions. In the next step, the configurations of feature points are evaluated using a novel criterion that is based on a Morphable Model and a combination of linear projections. Finally, the feature points initialize a model-fitting procedure of the Morphable Model. The result is a high-resolution 3D surface model.

PDF [BibTex]

PDF [BibTex]


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Mathematik der Wahrnehmung: Wendepunkte

Wichman, F., Ernst, MO.

Akademische Mitteilungen zw{\"o}lf: F{\"u}nf Sinne, pages: 32-37, 2007 (misc)

[BibTex]

[BibTex]


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Relative Entropy Policy Search

Peters, J.

CLMC Technical Report: TR-CLMC-2007-2, Computational Learning and Motor Control Lab, Los Angeles, CA, 2007, clmc (techreport)

Abstract
This technical report describes a cute idea of how to create new policy search approaches. It directly relates to the Natural Actor-Critic methods but allows the derivation of one shot solutions. Future work may include the application to interesting problems.

PDF link (url) [BibTex]

PDF link (url) [BibTex]

2004


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Fast Binary and Multi-Output Reduced Set Selection

Weston, J., Bakir, G.

(132), Max Planck Institute for Biological Cybernetics, Tübingen, Germany, November 2004 (techreport)

Abstract
We propose fast algorithms for reducing the number of kernel evaluations in the testing phase for methods such as Support Vector Machines (SVM) and Ridge Regression (RR). For non-sparse methods such as RR this results in significantly improved prediction time. For binary SVMs, which are already sparse in their expansion, the pay off is mainly in the cases of noisy or large-scale problems. However, we then further develop our method for multi-class problems where, after choosing the expansion to find vectors which describe all the hyperplanes jointly, we again achieve significant gains.

PostScript [BibTex]

2004

PostScript [BibTex]


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Joint Kernel Maps

Weston, J., Schölkopf, B., Bousquet, O., Mann, .., Noble, W.

(131), Max-Planck-Institute for Biological Cybernetics, Tübingen, November 2004 (techreport)

PDF [BibTex]

PDF [BibTex]


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Semi-Supervised Induction

Yu, K., Tresp, V., Zhou, D.

(141), Max Planck Institute for Biological Cybernetics, Tuebingen, Germany, August 2004 (techreport)

Abstract
Considerable progress was recently achieved on semi-supervised learning, which differs from the traditional supervised learning by additionally exploring the information of the unlabelled examples. However, a disadvantage of many existing methods is that it does not generalize to unseen inputs. This paper investigates learning methods that effectively make use of both labelled and unlabelled data to build predictive functions, which are defined on not just the seen inputs but the whole space. As a nice property, the proposed method allows effcient training and can easily handle new test points. We validate the method based on both toy data and real world data sets.

PDF PDF [BibTex]

PDF PDF [BibTex]


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On Hausdorff Distance Measures

Shapiro, MD., Blaschko, MB.

Department of Computer Science, University of Massachusetts Amherst, August 2004 (techreport)

[BibTex]

[BibTex]


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Object categorization with SVM: kernels for local features

Eichhorn, J., Chapelle, O.

(137), Max Planck Institute for Biological Cybernetics, Tübingen, Germany, July 2004 (techreport)

Abstract
In this paper, we propose to combine an efficient image representation based on local descriptors with a Support Vector Machine classifier in order to perform object categorization. For this purpose, we apply kernels defined on sets of vectors. After testing different combinations of kernel / local descriptors, we have been able to identify a very performant one.

PDF [BibTex]

PDF [BibTex]


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Hilbertian Metrics and Positive Definite Kernels on Probability Measures

Hein, M., Bousquet, O.

(126), Max Planck Institute for Biological Cybernetics, Tübingen, Germany, July 2004 (techreport)

Abstract
We investigate the problem of defining Hilbertian metrics resp. positive definite kernels on probability measures, continuing previous work. This type of kernels has shown very good results in text classification and has a wide range of possible applications. In this paper we extend the two-parameter family of Hilbertian metrics of Topsoe such that it now includes all commonly used Hilbertian metrics on probability measures. This allows us to do model selection among these metrics in an elegant and unified way. Second we investigate further our approach to incorporate similarity information of the probability space into the kernel. The analysis provides a better understanding of these kernels and gives in some cases a more efficient way to compute them. Finally we compare all proposed kernels in two text and one image classification problem.

PDF [BibTex]

PDF [BibTex]


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Kernels, Associated Structures and Generalizations

Hein, M., Bousquet, O.

(127), Max Planck Institute for Biological Cybernetics, Tübingen, Germany, July 2004 (techreport)

Abstract
This paper gives a survey of results in the mathematical literature on positive definite kernels and their associated structures. We concentrate on properties which seem potentially relevant for Machine Learning and try to clarify some results that have been misused in the literature. Moreover we consider different lines of generalizations of positive definite kernels. Namely we deal with operator-valued kernels and present the general framework of Hilbertian subspaces of Schwartz which we use to introduce kernels which are distributions. Finally indefinite kernels and their associated reproducing kernel spaces are considered.

PDF [BibTex]

PDF [BibTex]


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Triangle Fixing Algorithms for the Metric Nearness Problem

Dhillon, I., Sra, S., Tropp, J.

Univ. of Texas at Austin, June 2004 (techreport)

PDF [BibTex]

PDF [BibTex]


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Kamerakalibrierung und Tiefenschätzung: Ein Vergleich von klassischer Bündelblockausgleichung und statistischen Lernalgorithmen

Sinz, FH.

Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Tübingen, Germany, March 2004 (techreport)

Abstract
Die Arbeit verleicht zwei Herangehensweisen an das Problem der Sch{\"a}tzung der r{\"a}umliche Position eines Punktes aus den Bildkoordinaten in zwei verschiedenen Kameras. Die klassische Methode der B{\"u}ndelblockausgleichung modelliert zwei Einzelkameras und sch{\"a}tzt deren {\"a}ußere und innere Orientierung mit einer iterativen Kalibrationsmethode, deren Konvergenz sehr stark von guten Startwerten abh{\"a}ngt. Die Tiefensch{\"a}tzung eines Punkts geschieht durch die Invertierung von drei der insgesamt vier Projektionsgleichungen der Einzalkameramodelle. Die zweite Methode benutzt Kernel Ridge Regression und Support Vector Regression, um direkt eine Abbildung von den Bild- auf die Raumkoordinaten zu lernen. Die Resultate zeigen, daß der Ansatz mit maschinellem Lernen, neben einer erheblichen Vereinfachung des Kalibrationsprozesses, zu h{\"o}heren Positionsgenaugikeiten f{\"u}hren kann.

PDF [BibTex]

PDF [BibTex]


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Multivariate Regression with Stiefel Constraints

Bakir, G., Gretton, A., Franz, M., Schölkopf, B.

(128), MPI for Biological Cybernetics, Spemannstr 38, 72076, Tuebingen, 2004 (techreport)

Abstract
We introduce a new framework for regression between multi-dimensional spaces. Standard methods for solving this problem typically reduce the problem to one-dimensional regression by choosing features in the input and/or output spaces. These methods, which include PLS (partial least squares), KDE (kernel dependency estimation), and PCR (principal component regression), select features based on different a-priori judgments as to their relevance. Moreover, loss function and constraints are chosen not primarily on statistical grounds, but to simplify the resulting optimisation. By contrast, in our approach the feature construction and the regression estimation are performed jointly, directly minimizing a loss function that we specify, subject to a rank constraint. A major advantage of this approach is that the loss is no longer chosen according to the algorithmic requirements, but can be tailored to the characteristics of the task at hand; the features will then be optimal with respect to this objective. Our approach also allows for the possibility of using a regularizer in the optimization. Finally, by processing the observations sequentially, our algorithm is able to work on large scale problems.

PDF [BibTex]

PDF [BibTex]


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Learning from Labeled and Unlabeled Data Using Random Walks

Zhou, D., Schölkopf, B.

Max Planck Institute for Biological Cybernetics, 2004 (techreport)

Abstract
We consider the general problem of learning from labeled and unlabeled data. Given a set of points, some of them are labeled, and the remaining points are unlabeled. The goal is to predict the labels of the unlabeled points. Any supervised learning algorithm can be applied to this problem, for instance, Support Vector Machines (SVMs). The problem of our interest is if we can implement a classifier which uses the unlabeled data information in some way and has higher accuracy than the classifiers which use the labeled data only. Recently we proposed a simple algorithm, which can substantially benefit from large amounts of unlabeled data and demonstrates clear superiority to supervised learning methods. In this paper we further investigate the algorithm using random walks and spectral graph theory, which shed light on the key steps in this algorithm.

PDF PostScript [BibTex]

PDF PostScript [BibTex]


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Behaviour and Convergence of the Constrained Covariance

Gretton, A., Smola, A., Bousquet, O., Herbrich, R., Schölkopf, B., Logothetis, N.

(130), MPI for Biological Cybernetics, 2004 (techreport)

Abstract
We discuss reproducing kernel Hilbert space (RKHS)-based measures of statistical dependence, with emphasis on constrained covariance (COCO), a novel criterion to test dependence of random variables. We show that COCO is a test for independence if and only if the associated RKHSs are universal. That said, no independence test exists that can distinguish dependent and independent random variables in all circumstances. Dependent random variables can result in a COCO which is arbitrarily close to zero when the source densities are highly non-smooth, which can make dependence hard to detect empirically. All current kernel-based independence tests share this behaviour. Finally, we demonstrate exponential convergence between the population and empirical COCO, which implies that COCO does not suffer from slow learning rates when used as a dependence test.

PDF [BibTex]

PDF [BibTex]


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Statistische Lerntheorie und Empirische Inferenz

Schölkopf, B.

Jahrbuch der Max-Planck-Gesellschaft, 2004, pages: 377-382, 2004 (misc)

Abstract
Statistical learning theory studies the process of inferring regularities from empirical data. The fundamental problem is what is called generalization: how it is possible to infer a law which will be valid for an infinite number of future observations, given only a finite amount of data? This problem hinges upon fundamental issues of statistics and science in general, such as the problems of complexity of explanations, a priori knowledge, and representation of data.

PDF Web [BibTex]

PDF Web [BibTex]


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Confidence Sets for Ratios: A Purely Geometric Approach To Fieller’s Theorem

von Luxburg, U., Franz, V.

(133), Max Planck Institute for Biological Cybernetics, 2004 (techreport)

Abstract
We present a simple, geometric method to construct Fieller's exact confidence sets for ratios of jointly normally distributed random variables. Contrary to previous geometric approaches in the literature, our method is valid in the general case where both sample mean and covariance are unknown. Moreover, not only the construction but also its proof are purely geometric and elementary, thus giving intuition into the nature of the confidence sets.

PDF [BibTex]

PDF [BibTex]


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Transductive Inference with Graphs

Zhou, D., Schölkopf, B.

Max Planck Institute for Biological Cybernetics, 2004, See the improved version Regularization on Discrete Spaces. (techreport)

Abstract
We propose a general regularization framework for transductive inference. The given data are thought of as a graph, where the edges encode the pairwise relationships among data. We develop discrete analysis and geometry on graphs, and then naturally adapt the classical regularization in the continuous case to the graph situation. A new and effective algorithm is derived from this general framework, as well as an approach we developed before.

[BibTex]

[BibTex]


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Kompetenzerwerb für Informationssysteme - Einfluss des Lernprozesses auf die Interaktion mit Fahrerinformationssystemen. Veröffentlichter Abschlussbericht (Förderkennzeichen BaSt FE 82.196/2001).

Totzke, I., Krüger, H., Hofmann, M., Meilinger, T., Rauch, N., Schmidt, G.

Interdisziplinäres Zentrum für Verkehrswissenschaften (IZVW), Würzburg, 2004 (techreport)

[BibTex]

[BibTex]

2002


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Kernel Dependency Estimation

Weston, J., Chapelle, O., Elisseeff, A., Schölkopf, B., Vapnik, V.

(98), Max Planck Institute for Biological Cybernetics, August 2002 (techreport)

Abstract
We consider the learning problem of finding a dependency between a general class of objects and another, possibly different, general class of objects. The objects can be for example: vectors, images, strings, trees or graphs. Such a task is made possible by employing similarity measures in both input and output spaces using kernel functions, thus embedding the objects into vector spaces. Output kernels also make it possible to encode prior information and/or invariances in the loss function in an elegant way. We experimentally validate our approach on several tasks: mapping strings to strings, pattern recognition, and reconstruction from partial images.

PDF [BibTex]

2002

PDF [BibTex]


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Global Geometry of SVM Classifiers

Zhou, D., Xiao, B., Zhou, H., Dai, R.

Max Planck Institute for Biological Cybernetics, Tübingen, Germany, June 2002 (techreport)

Abstract
We construct an geometry framework for any norm Support Vector Machine (SVM) classifiers. Within this framework, separating hyperplanes, dual descriptions and solutions of SVM classifiers are constructed by a purely geometric fashion. In contrast with the optimization theory used in SVM classifiers, we have no complicated computations any more. Each step in our theory is guided by elegant geometric intuitions.

PDF PostScript [BibTex]

PDF PostScript [BibTex]


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Computationally Efficient Face Detection

Romdhani, S., Torr, P., Schölkopf, B., Blake, A.

(MSR-TR-2002-69), Microsoft Research, June 2002 (techreport)

Web [BibTex]

Web [BibTex]


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Kernel-based nonlinear blind source separation

Harmeling, S., Ziehe, A., Kawanabe, M., Müller, K.

EU-Project BLISS, January 2002 (techreport)

GZIP [BibTex]

GZIP [BibTex]


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A compression approach to support vector model selection

von Luxburg, U., Bousquet, O., Schölkopf, B.

(101), Max Planck Institute for Biological Cybernetics, 2002, see more detailed JMLR version (techreport)

Abstract
In this paper we investigate connections between statistical learning theory and data compression on the basis of support vector machine (SVM) model selection. Inspired by several generalization bounds we construct ``compression coefficients'' for SVMs, which measure the amount by which the training labels can be compressed by some classification hypothesis. The main idea is to relate the coding precision of this hypothesis to the width of the margin of the SVM. The compression coefficients connect well known quantities such as the radius-margin ratio R^2/rho^2, the eigenvalues of the kernel matrix and the number of support vectors. To test whether they are useful in practice we ran model selection experiments on several real world datasets. As a result we found that compression coefficients can fairly accurately predict the parameters for which the test error is minimized.

[BibTex]

[BibTex]


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Feature Selection and Transduction for Prediction of Molecular Bioactivity for Drug Design

Weston, J., Perez-Cruz, F., Bousquet, O., Chapelle, O., Elisseeff, A., Schölkopf, B.

Max Planck Institute for Biological Cybernetics / Biowulf Technologies, 2002 (techreport)

Web [BibTex]

Web [BibTex]


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Observations on the Nyström Method for Gaussian Process Prediction

Williams, C., Rasmussen, C., Schwaighofer, A., Tresp, V.

Max Planck Institute for Biological Cybernetics, Tübingen, Germany, 2002 (techreport)

Abstract
A number of methods for speeding up Gaussian Process (GP) prediction have been proposed, including the Nystr{\"o}m method of Williams and Seeger (2001). In this paper we focus on two issues (1) the relationship of the Nystr{\"o}m method to the Subset of Regressors method (Poggio and Girosi 1990; Luo and Wahba, 1997) and (2) understanding in what circumstances the Nystr{\"o}m approximation would be expected to provide a good approximation to exact GP regression.

PostScript [BibTex]

PostScript [BibTex]

1999


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Estimating the support of a high-dimensional distribution

Schölkopf, B., Platt, J., Shawe-Taylor, J., Smola, A., Williamson, R.

(MSR-TR-99-87), Microsoft Research, 1999 (techreport)

Web [BibTex]

1999

Web [BibTex]


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Generalization Bounds via Eigenvalues of the Gram matrix

Schölkopf, B., Shawe-Taylor, J., Smola, A., Williamson, R.

(99-035), NeuroCOLT, 1999 (techreport)

[BibTex]

[BibTex]


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Sparse kernel feature analysis

Smola, A., Mangasarian, O., Schölkopf, B.

(99-04), Data Mining Institute, 1999, 24th Annual Conference of Gesellschaft f{\"u}r Klassifikation, University of Passau (techreport)

PostScript [BibTex]

PostScript [BibTex]