We describe methods for computing an implicit model of a hypersurface that is given only by a finite sampling. The methods work by mapping the sample points into a reproducing kernel Hilbert space and then determining regions in terms of hyperplanes.

We describe semi-Markov conditional random fields (semi-CRFs), a conditionally trained version of semi-Markov chains. Intuitively, a semi-CRF on an input sequence x outputs a "segmentation" of x, in which labels are assigned to segments (i.e., subsequences) of x rather than to individual elements x

We present a semi-parametric latent variable model based technique for density modelling, dimensionality reduction and visualization. Unlike previous methods, we estimate the latent distribution non-parametrically which enables us to model data generated by an underlying low dimensional, multimodal distribution. In addition, we allow the components of latent variable models to be drawn from the exponential family which makes the method suitable for special data types, for example binary or count data. Simulations on real valued, binary and count data show favorable comparison to other related schemes both in terms of separating different populations and generalization to unseen samples.

The problem of detecting "atypical objects" or "outliers" is one of the classical topics in (robust) statistics. Recently, it has been proposed to address this problem by means of one-class SVM classifiers. The main conceptual shortcoming of most one-class approaches, however, is that in a strict sense they are unable to detect outliers, since the expected fraction of outliers has to be specified in advance. The method presented in this paper overcomes this problem by relating kernelized one-class classification to Gaussian density estimation in the induced feature space. Having established this relation, it is possible to identify "atypical objects" by quantifying their deviations from the Gaussian model. For RBF kernels it is shown that the Gaussian model is "rich enough" in the sense that it asymptotically provides an unbiased estimator for the true density. In order to overcome the inherent model selection problem, a cross-validated likelihood criterion for selecting all free model parameters is applied.

Regularization plays a central role in the analysis of modern data, where non-regularized fitting is likely to lead to over-fitted models, useless for both prediction and interpretation. We consider the design of incremental algorithms which follow paths of regularized solutions, as the regularization varies. These approaches often result in methods which are both efficient and highly flexible. We suggest a general path-following algorithm based on second-order approximations, prove that under mild conditions it remains "very close" to the path of optimal solutions and illustrate it with examples.

Many works have shown that strong connections relate learning from examples to regularization techniques for ill-posed inverse problems. Nevertheless by now there was no formal evidence neither that learning from examples could be seen as an inverse problem nor that theoretical results in learning theory could be independently derived using tools from regularization theory. In this paper we provide a positive answer to both questions. Indeed, considering the square loss, we translate the learning problem in the language of regularization theory and show that consistency results and optimal regularization parameter choice can be derived by the discretization of the corresponding inverse problem.

We introduce a novel active-learning scenario in which a user wants to work with a learning algorithm to identify useful anomalies. These are distinguished from the traditional statistical definition of anomalies as outliers or merely ill-modeled points. Our distinction is that the usefulness of anomalies is categorized subjectively by the user. We make two additional assumptions. First, there exist extremely few useful anomalies to be hunted down within a massive dataset.

A novel linear feature selection algorithm is presented based on the global minimization of a data-dependent generalization error bound. Feature selection and scaling algorithms often lead to non-convex optimization problems, which in many previous approaches were addressed through gradient descent procedures that can only guarantee convergence to a local minimum. We propose an alternative approach, whereby the global solution of the non-convex optimization problem is derived via an equivalent optimization problem. Moreover, the convex optimization task is reduced to a conic quadratic programming problem for which efficient solvers are available. Highly competitive numerical results on both artificial and real-world data sets are reported.

Dominant sets are a new graph-theoretic concept that has proven to be relevant in pairwise data clustering problems, such as image segmentation. They generalize the notion of a maximal clique to edgeweighted graphs and have intriguing, nontrivial connections to continuous quadratic optimization and spectral-based grouping. We address the problem of grouping out-of-sample examples after the clustering process has taken place. This may serve either to drastically reduce the computational burden associated to the processing of very large data sets, or to efficiently deal with dynamic situations whereby data sets need to be updated continually. We show that the very notion of a dominant set offers a simple and efficient way of doing this. Numerical experiments on various grouping problems show the effectiveness of the approach.

Capturing dependencies in images in an unsupervised manner is important for many image processing applications. We propose a new method for capturing nonlinear dependencies in images of natural scenes. This method is an extension of the linear Independent Component Analysis (ICA) method by building a hierarchical model based on ICA and mixture of Laplacian distribution. The model parameters are learned via an EM algorithm and it can accurately capture variance correlation and other high order structures in a simple manner. We visualize the learned variance structure and demonstrate applications to image segmentation and denoising.