We consider online learning in a Reproducing Kernel Hilbert Space. Our method is computationally efficient and leads to simple algorithms. In particular we derive update equations for classification, regression, and novelty detection. The inclusion of the -trick allows us to give a robust parameterization.

In classical large information retrieval systems, the system responds to a user initiated query with a list of results ranked by relevance. The users may further refine their query as needed. This process may result in a lengthy correspondence without conclusion. We propose an alternative active learning approach, where the system responds to the initial user's query by successively probing the user for distinctions at multiple levels of abstraction. The system's initiated queries are optimized for speedy recovery and the user is permitted to respond with multiple selections or may reject the query. The information is in each case unambiguously incorporated by the system and the subsequent queries are adjusted to minimize the need for further exchange. The system's initiated queries are subject to resource constraints pertaining to the amount of information that can be presented to the user per iteration.

We propose a novel clustering method that is an extension of ideas inherent to scale-space clustering and support-vector clustering. Like the latter, it associates 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 probability function. 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.

In kernel based learning the data is mapped to a kernel feature space of a dimension that corresponds to the number of training data points. In practice, however, the data forms a smaller submanifold in feature space, a fact that has been used e.g. by reduced set techniques for SVMs. We propose a new mathematical construction that permits to adapt to the intrinsic dimension and to find an orthonormal basis of this submanifold. In doing so, computations get much simpler and more important our theoretical framework allows to derive elegant kernelized blind source separation (BSS) algorithms for arbitrary invertible nonlinear mixings. Experiments demonstrate the good performance and high computational efficiency of our kTDSEP algorithm for the problem of nonlinear BSS.

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 the intra-class disparity.

In many scientific and engineering applications, detecting and understanding differences between two groups of examples can be reduced to a classical problem of training a classifier for labeling new examples while making as few mistakes as possible. In the traditional classification setting, the resulting classifier is rarely analyzed in terms of the properties of the input data captured by the discriminative model. However, such analysis is crucial if we want to understand and visualize the detected differences. We propose an approach to interpretation of the statistical model in the original feature space that allows us to argue about the model in terms of the relevant changes to the input vectors. For each point in the input space, we define a discriminative direction to be the direction that moves the point towards the other class while introducing as little irrelevant change as possible with respect to the classifier function. We derive the discriminative direction for kernel-based classifiers, demonstrate the technique on several examples and briefly discuss its use in the statistical shape analysis, an application that originally motivated this work.

Since the discovery that the best error-correcting decoding algorithm can be viewed as belief propagation in a cycle-bound graph, researchers have been trying to determine under what circumstances "loopy belief propagation" is effective for probabilistic inference. Despite several theoretical advances in our understanding of loopy belief propagation, to our knowledge, the only problem that has been solved using loopy belief propagation is error-correcting decoding on Gaussian channels. We propose a new representation for the two-dimensional phase unwrapping problem, and we show that loopy belief propagation produces results that are superior to existing techniques. This is an important result, since many imaging techniques, including magnetic resonance imaging and interferometric synthetic aperture radar, produce phase-wrapped images. Interestingly, the graph that we use has a very large number of very short cycles, supporting evidence that a large minimum cycle length is not needed for excellent results using belief propagation. 1 Introduction Phase unwrapping is an easily stated, fundamental problem in image processing (Ghiglia and Pritt 1998).

Factor analysis and principal components analysis can be used to model linear relationships between observed variables and linearly map high-dimensional data to a lower-dimensional hidden space. In factor analysis, the observations are modeled as a linear combination of normally distributed hidden variables. We describe a nonlinear generalization of factor analysis, called "product analysis", that models the observed variables as a linear combination of products of normally distributed hidden variables. Just as factor analysis can be viewed as unsupervised linear regression on unobserved, normally distributed hidden variables, product analysis can be viewed as unsupervised linear regression on products of unobserved, normally distributed hidden variables. The mapping between the data and the hidden space is nonlinear, so we use an approximate variational technique for inference and learning.

In previous work on "transformed mixtures of Gaussians" and "transformed hidden Markov models", we showed how the EM algorithm in a discrete latent variable model can be used to jointly normalize data (e.g., center images, pitch-normalize spectrograms) and learn a mixture model of the normalized data. The only input to the algorithm is the data, a list of possible transformations, and the number of clusters to find. The main criticism of this work was that the exhaustive computation of the posterior probabilities over transformations would make scaling up to large feature vectors and large sets of transformations intractable. Here, we describe how a tremendous speedup is acheived through the use of a variational technique for decoupling transformations, and a fast Fourier transform method for computing posterior probabilities.

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.