We extend the well-known BFGS quasi-Newton method and its limited-memory variant LBFGS to the optimization of non-smooth convex objectives. This is done in a rigorous fashion by generalizing three components of BFGS to subdifferentials: The local quadratic model, the identification of a descent direction, and the Wolfe line search conditions. We apply the resulting subLBFGS algorithm to L2-regularized risk minimization with binary hinge loss, and its direction-finding component to L1-regularized risk minimization with logistic loss. In both settings our generic algorithms perform comparable to or better than their counterparts in specialized state-of-the-art solvers.

BACKGROUND: When analysing microarray and other small sample size biological datasets, care is needed to avoid various biases. We analyse a form of bias, stratification bias, that can substantially affect analyses using sample-reuse validation techniques and lead to inaccurate results. This bias is due to imperfect stratification of samples in the training and test sets and the dependency between these stratification errors, i.e. the variations in class proportions in the training and test sets are negatively correlated. RESULTS: We show that when estimating the performance of classifiers on low signal datasets (i.e. those which are difficult to classify), which are typical of many prognostic microarray studies, commonly used performance measures can suffer from a substantial negative bias. For error rate this bias is only severe in quite restricted situations, but can be much larger and more frequent when using ranking measures such as the receiver operating characteristic (ROC) curve and area under the ROC (AUC). Substantial biases are shown in simulations and on the van 't Veer breast cancer dataset. The classification error rate can have large negative biases for balanced datasets, whereas the AUC shows substantial pessimistic biases even for imbalanced datasets. In simulation studies using 10-fold cross-validation, AUC values of less than 0.3 can be observed on random datasets rather than the expected 0.5. Further experiments on the van 't Veer breast cancer dataset show these biases exist in practice. CONCLUSION: Stratification bias can substantially affect several performance measures. In computing the AUC, the strategy of pooling the test samples from the various folds of cross-validation can lead to large biases; computing it as the average of per-fold estimates avoids this bias and is thus the recommended approach. As a more general solution applicable to other performance measures, we show that stratified repeated holdout and a modified version of k-fold cross-validation, balanced, stratified cross-validation and balanced leave-one-out cross-validation, avoids the bias. Therefore for model selection and evaluation of microarray and other small biological datasets, these methods should be used and unstratified versions avoided. In particular, the commonly used (unbalanced) leave-one-out cross-validation should not be used to estimate AUC for small datasets.

We introduce two methods to improve convergence of the Kernel Hebbian Algorithm (KHA) for iterative kernel PCA. KHA has a scalar gain parameter which is either held constant or decreased as 1/t, leading to slow convergence. Our KHA/et algorithm accelerates KHA by incorporating the reciprocal of the current estimated eigenvalues as a gain vector. We then derive and apply Stochastic MetaDescent (SMD) to KHA/et; this further speeds convergence by performing gain adaptation in RKHS. Experimental results for kernel PCA and spectral clustering of USPS digits as well as motion capture and image de-noising problems confirm that our methods converge substantially faster than conventional KHA.

Handwritten text recognition is one of the most difficult problems in the field of pattern recognition. In this paper, we describe our efforts towards improving the performance of state-of-the-art handwriting recognition systems through the use of classifier ensembles. There are many examples of classification problems in the literature where multiple classifier systems increase the performance over single classifiers. Normally one of the two following approaches is used to create a multiple classifier system. (1) Several classifiers are developed completely independent of each other and combined in a last step. (2) Several classifiers are created out of one prototype classifier by using so-called classifier ensemble creation methods. In this paper an algorithm which combines both approaches is introduced and it is used to increase the recognition rate of a hidden Markov model (HMM) based handwritten word recognizer.

Handwritten text recognition is one of the most difficult problems in the field of pattern recognition. Recently, a number of classifier creation methods, known as ensemble methods, have been proposed in the field of machine learning. It has been shown that these methods are able to substantially improve recognition performance in complex classification tasks. In this paper we examine the influence of the vocabulary size and the number of training samples on the performance of three ensemble methods in the context of handwritten word recognition. The experiments were conducted with two different offline hidden Markov model based handwritten word recognizers.

In off-line handwriting recognition, classifiers based on hidden Markov models (HMMs) have become very popular. However, while there exist well-established training algorithms, such as the Baum-Welsh procedure, which optimize the transition and output probabilities of a given HMM architecture, the architecture itself, and in particular the number of states, must be chosen "by hand". Also the number of training iterations and the output distributions need to be defined by the system designer. In this paper we examine some optimization strategies for an HMM classifier that works with continuous feature values and uses the Baum-Welch training algorithm. The free parameters of the optimization procedure introduced in this paper are the number of states of a model, the number of training iterations, and the number of Gaussian mixtures for each state. The proposed optimization strategies are evaluated in the context of a handwritten word recognition task.