We develop an intuitive geometric framework for support vector regression (SVR). By examining when ɛ-tubes exist, we show that SVR can be regarded as a classification problem in the dual space. Hard and soft ɛ-tubes are constructed by separating the convex or reduced convex hulls respectively of the training data with the response variable shifted up and down by ɛ. A novel SVR model is proposed based on choosing the max-margin plane between the two shifted datasets.

The support vector machine (SVM) is known for its good performance in binary classification, but its extension to multi-class classification is still an ongoing research issue. In this paper, we propose a new approach for classification, called the import vector machine (IVM), which is built on kernel logistic regression (KLR). We show that the IVM not only performs as well as the SVM in binary classification, but also can naturally be generalized to the multi-class case. Furthermore, the IVM provides an estimate of the underlying probability. Similar to the "support points" of the SVM, the IVM model uses only a fraction of the training data to index kernel basis functions, typically a much smaller fraction than the SVM. This gives the IVM a computational advantage over the SVM, especially when the size of the training data set is large.

We give an unified convergence analysis of ensemble learning methods including e.g. AdaBoost, Logistic Regression and the Least-Square- Boost algorithm for regression. These methods have in common that they iteratively call a base learning algorithm which returns hypotheses that are then linearly combined. We show that these methods are related to the Gauss-Southwell method known from numerical optimization and state non-asymptotical convergence results for all these methods. Our analysis includes -norm regularized cost functions leading to a clean and general way to regularize ensemble learning.

Discriminative classifiers model the posterior p(ylx) directly, or learn a direct map from inputs x to the class labels. There are several compelling reasons for using discriminative rather than generative classifiers, one of which, succinctly articulated by Vapnik [6], is that "one should solve the [classification] problem directly and never solve a more general problem as an intermediate step [such as modeling p(xly)]." Indeed, leaving aside computational issues and matters such as handling missing data, the prevailing consensus seems to be that discriminative classifiers are almost always to be preferred to generative ones. Another piece of prevailing folk wisdom is that the number of examples needed to fit a model is often roughly linear in the number of free parameters of a model. This has its theoretical basis in the observation that for "many" models, the VC dimension is roughly linear or at most some low-order polynomial in the number of parameters (see, e.g., [1, 3]), and it is known that sample complexity in the discriminative setting is linear in the VC dimension [6]. In this paper, we study empirically and theoretically the extent to which these beliefs are true. A parametric family of probabilistic models p(x, y) can be fit either to optimize the joint likelihood of the inputs and the labels, or fit to optimize the conditional likelihood p(ylx), or even fit to minimize the 0-1 training error obtained by thresholding p(ylx) to make predictions.

We develop an intuitive geometric framework for support vector regression (SVR). By examining when ɛ-tubes exist, we show that SVR can be regarded as a classification problem in the dual space. Hard and soft ɛ-tubes are constructed by separating the convex or reduced convex hulls respectively of the training data with the response variable shifted up and down by ɛ. A novel SVR model is proposed based on choosing the max-margin plane between the two shifted datasets.

The support vector machine (SVM) is known for its good performance in binary classification, but its extension to multi-class classification is still an ongoing research issue. In this paper, we propose a new approach for classification, called the import vector machine (IVM), which is built on kernel logistic regression (KLR). We show that the IVM not only performs as well as the SVM in binary classification, but also can naturally be generalized to the multi-class case. Furthermore, the IVM provides an estimate of the underlying probability. Similar to the "support points" of the SVM, the IVM model uses only a fraction of the training data to index kernel basis functions, typically a much smaller fraction than the SVM. This gives the IVM a computational advantage over the SVM, especially when the size of the training data set is large.

We present an extension to the Mixture of Experts (ME) model, where the individual experts are Gaussian Process (GP) regression models. Using an input-dependent adaptation of the Dirichlet Process, we implement a gating network for an infinite number of Experts. Inference in this model may be done efficiently using a Markov Chain relying on Gibbs sampling. The model allows the effective covariance function to vary with the inputs, and may handle large datasets - thus potentially overcoming two of the biggest hurdles with GP models.

We develop an intuitive geometric framework for support vector regression (SVR). By examining when ɛ-tubes exist, we show that SVR can be regarded as a classification problem in the dual space. Hard and soft ɛ-tubes are constructed by separating the convex or reduced convex hulls respectively of the training data with the response variable shifted up and down by ɛ. A novel SVR model is proposed based on choosing the max-margin plane between the two shifted datasets.

We present an extension to the Mixture of Experts (ME) model, where the individual experts are Gaussian Process (GP) regression models. Using aninput-dependent adaptation of the Dirichlet Process, we implement agating network for an infinite number of Experts. Inference in this model may be done efficiently using a Markov Chain relying on Gibbs sampling. The model allows the effective covariance function to vary with the inputs, and may handle large datasets - thus potentially overcoming twoof the biggest hurdles with GP models.

We give an unified convergence analysis of ensemble learning methods includinge.g. AdaBoost, Logistic Regression and the Least-Square- Boost algorithm for regression. These methods have in common that they iteratively call a base learning algorithm which returns hypotheses that are then linearly combined. We show that these methods are related to the Gauss-Southwell method known from numerical optimization and state non-asymptotical convergence results for all these methods. Our analysis includes -norm regularized cost functions leading to a clean and general way to regularize ensemble learning.