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.

Recently, Jaakkola and Haussler proposed a method for constructing kernelfunctions from probabilistic models. Their so called "Fisher kernel" has been combined with discriminative classifiers such as SVM and applied successfully in e.g.

Drawing on the correspondence between the graph Laplacian, the Laplace-Beltrami operator on a manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for constructing a representation for data sampled from a low dimensional manifoldembedded in a higher dimensional space. The algorithm provides a computationally efficient approach to nonlinear dimensionalityreduction that has locality preserving properties and a natural connection to clustering.

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.

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This paper presents an unsupervised learning algorithm that can derive the probabilistic dependence structure of parts of an object (a moving human bodyin our examples) automatically from unlabeled data. The distinguished partof this work is that it is based on unlabeled data, i.e., the training features include both useful foreground parts and background clutter and the correspondence between the parts and detected features are unknown. We use decomposable triangulated graphs to depict the probabilistic independence of parts, but the unsupervised technique is not limited to this type of graph. In the new approach, labeling of the data (part assignments) is taken as hidden variables and the EM algorithm isapplied. A greedy algorithm is developed to select parts and to search for the optimal structure based on the differential entropy of these variables. The success of our algorithm is demonstrated by applying it to generate models of human motion automatically from unlabeled real image sequences.

We describe a computer system that provides a real-time musical accompanimentfor a live soloist in a piece of non-improvised music for soloist and accompaniment. A Bayesian network is developed thatrepresents the joint distribution on the times at which the solo and accompaniment notes are played, relating the two parts through a layer of hidden variables. The network is first constructed usingthe rhythmic information contained in the musical score. The network is then trained to capture the musical interpretations ofthe soloist and accompanist in an off-line rehearsal phase. During live accompaniment the learned distribution of the network is combined with a real-time analysis of the soloist's acoustic signal, performedwith a hidden Markov model, to generate a musically principledaccompaniment that respects all available sources of knowledge. A live demonstration will be provided.

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, oneof 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. Anotherpiece 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.

It has been noted by several authors that Belief Propagation can can also give impressive results for graphs that are not trees [2]. The Cluster Variation Method (CVM), is a method that has been developed in the physics community for approximate inference in the Ising model [3]. The CVM approximates thejoint probability distribution by a number of (overlapping) marginal distributions (clusters). The quality of the approximation is determined by the size and number of clusters. When the clusters consist of only two variables, the method is known as the Bethe approximation.