We investigate a learning algorithm for the classification of nonnegative data by mixture models. Multiplicative update rules are derived that directly optimize the performance of these models as classifiers. The update rules have a simple closed form and an intuitive appeal. Our algorithm retains the main virtues of the Expectation-Maximization (EM) algorithm--its guarantee of monotonic improvement, andits absence of tuning parameters--with the added advantage of optimizing a discriminative objective function. The algorithm reduces as a special caseto the method of generalized iterative scaling for log-linear models. The learning rate of the algorithm is controlled by the sparseness of the training data. We use the method of nonnegative matrix factorization (NMF) to discover sparse distributed representations of the data. This form of feature selection greatly accelerates learning and makes the algorithm practical on large problems. Experiments showthat discriminatively trained mixture models lead to much better classification than comparably sized models trained by EM.

We suggest a nonparametric framework for unsupervised learning of projection models in terms of density estimation on quantized sample spaces. The objective is not to optimally reconstruct the data but instead thequantizer is chosen to optimally reconstruct the density of the data. For the resulting quantizing density estimator (QDE) we present a general method for parameter estimation and model selection. We show how projection sets which correspond to traditional unsupervised methods likevector quantization or PCA appear in the new framework. For a principal component quantizer we present results on synthetic and realworld data,which show that the QDE can improve the generalization of the kernel density estimator although its estimate is based on significantly lower-dimensional projection indices of the data.

We propose a novel clustering method that is an extension of ideas inherent toscale-space clustering and support-vector clustering. Like the latter, itassociates every data point with a vector in Hilbert space, and like the former it puts emphasis on their total sum, that is equal to the scalespace probabilityfunction. The novelty of our approach is the study of an operator in Hilbert space, represented by the Schrödinger equation of which the probability function is a solution. This Schrödinger equation contains a potential function that can be derived analytically from the probability function.

Maximum margin classifiers such as Support Vector Machines (SVMs) critically depends upon the convex hulls of the training samples of each class, as they implicitly search for the minimum distance between the convex hulls. We propose Extrapolated Vector Machines(XVMs) which rely on extrapolations outside these convex hulls. XVMs improve SVM generalization very significantly on the MNIST [7] OCR data. They share similarities with the Fisher discriminant: maximize the inter-class margin while minimizing theintra-class disparity.

This conditional expected claim amount is called the pure premium and it is the basis of the gross premium charged to the insured. This expected value is conditionned on information available about the insured and about the contract, which we call input profile here. This regression problem is difficult for several reasons: large number of examples, -large number variables (most of which are discrete and multi-valued), non-stationarity of the distribution, and a conditional distribution of the dependent variable which is very different from those usually encountered in typical applications .of

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 aswell 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.

The choice of an SVM kernel corresponds to the choice of a representation ofthe data in a feature space and, to improve performance, it should therefore incorporate prior knowledge such as known transformation invariances. We propose a technique which extends earlier work and aims at incorporating invariances in nonlinear kernels.We show on a digit recognition task that the proposed approach is superior to the Virtual Support Vector method, which previously had been the method of choice. 1 Introduction In some classification tasks, an a priori knowledge is known about the invariances related to the task. For instance, in image classification, we know that the label of a given image should not change after a small translation or rotation.

Support Vector Machines (SVMs) are currently the state-of-the-art models for many classification problems but they suffer from the complexity of their training algorithmwhich is at least quadratic with respect to the number of examples. Hence, it is hopeless to try to solve real-life problems having more than a few hundreds of thousands examples with SVMs. The present paper proposes a new mixture of SVMs that can be easily implemented in parallel and where each SVM is trained on a small subset of the whole dataset. Experiments on a large benchmark dataset (Forest) as well as a difficult speech database, yielded significant time improvement (time complexity appears empirically to locally grow linearly with the number of examples) . In addition, and that is a surprise, a significant improvement in generalization was observed on Forest. 1 Introduction Recently a lot of work has been done around Support Vector Machines [9], mainly due to their impressive generalization performances on classification problems when compared to other algorithms such as artificial neural networks [3, 6].

We give results about the learnability and required complexity of logical formulae to solve classification problems. These results are obtained by linking propositional logic with kernel machines. In particular we show that decision trees and disjunctive normal forms (DNF) can be represented bythe help of a special kernel, linking regularized risk to separation margin. Subsequently we derive a number of lower bounds on the required complexity of logic formulae using properties of algorithms for generation of linear estimators, such as perceptron and maximal perceptron learning.

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