Locally Linear Embedding (LLE) is an elegant nonlinear dimensionality-reduction technique recently introduced by Roweis and Saul [2]. It fails when the data is divided into separate groups. We study a variant of LLE that can simultaneously group the data and calculate local embedding of each group. An estimate for the upper bound on the intrinsic dimension of the data set is obtained automatically. 1 Introduction

In this paper we show that online algorithms for classification and regression canbe naturally used to obtain hypotheses with good datadependent tailbounds on their risk. Our results are proven without requiring complicated concentration-of-measure arguments and they hold for arbitrary online learning algorithms. Furthermore, when applied to concrete online algorithms, our results yield tail bounds that in many cases are comparable or better than the best known bounds.

We propose randomized techniques for speeding up Kernel Principal Component Analysis on three levels: sampling and quantization of the Gram matrix in training, randomized rounding in evaluating the kernel expansions, and random projections in evaluating the kernel itself. In all three cases, we give sharp bounds on the accuracy of the obtained approximations. Ratherintriguingly, all three techniques can be viewed as instantiations of the following idea: replace the kernel function by a "randomized kernel" which behaves like in expectation.

Over the last years, particle filters have been applied with great success to a variety of state estimation problems. We present a statistical approach to increasing the efficiency of particle filters by adapting the size of sample sets on-the-fly. The key idea of the KLD-sampling method is to bound the approximation error introduced by the sample-based representation of the particle filter. The name KLD-sampling is due to the fact that we measure the approximation error by the Kullback-Leibler distance. Our adaptation approach chooses a small number of samples if the density is focused on a small part of the state space, and it chooses a large number of samples if the state uncertainty is high. Both the implementation and computation overhead of this approach are small. Extensive experiments using mobile robot localization as a test application show that our approach yields drastic improvements over particle filters with fixed sample set sizes and over a previously introduced adaptation technique.

We propose a framework based on a parametric quadratic programming (QP)technique to solve the support vector machine (SVM) training problem. This framework, can be specialized to obtain two SVM optimization methods. The first solves the fixed bias problem, whilethe second starts with an optimal solution for a fixed bias problem and adjusts the bias until the optimal value is found. The later method can be applied in conjunction with any other existing techniquewhich obtains a fixed bias solution. Moreover, the second method can also be used independently to solve the complete SVMtraining problem. A combination of these two methods is more flexible than each individual method and, among other things, produces an incremental algorithm which exactly solve the 1-Norm Soft Margin SVM optimization problem. Applying Selective Samplingtechniques may further boost convergence.

Guided by an initial idea of building a complex (non linear) decision surface with maximal local margin in input space, we give a possible geometrical intuition as to why K-Nearest Neighbor (KNN) algorithms often perform more poorly than SVMs on classification tasks. We then propose modified K-Nearest Neighbor algorithms to overcome the perceived problem.The approach is similar in spirit to Tangent Distance, but with invariances inferred from the local neighborhood rather than prior knowledge. Experimental results on real world classification tasks suggest thatthe modified KNN algorithms often give a dramatic improvement overstandard KNN and perform as well or better than SVMs.

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.

We investigate the generalization performance of some learning problems inHilbert functional Spaces. We introduce a notion of convergence of the estimated functional predictor to the best underlying predictor, and obtain an estimate on the rate of the convergence. This estimate allows us to derive generalization bounds on some learning formulations.

The information bottleneck method is an unsupervised model independent data organization technique. Given a joint distribution peA, B), this method constructs anew variable T that extracts partitions, or clusters, over the values of A that are informative about B. In a recent paper, we introduced a general principled frameworkfor multivariate extensions of the information bottleneck method that allows us to consider multiple systems of data partitions that are interrelated. In this paper, we present a new family of simple agglomerative algorithms to construct such systems of interrelated clusters. We analyze the behavior of these algorithms and apply them to several real-life datasets.

Our BFBP (Batch Fit to Best Paths) algorithm alternates between an exploration phase (during which trajectories are generated to try to find fragments of the optimal policy) and a function fitting phase (during which a function approximator is fit to the best known paths from start states to terminal states). An advantage of this approach is that batch value-function fitting is a global process, which allows it to address the tradeoffs in function approximation that cannot be handled by local, online algorithms.