Bayesian inference has become increasingly important in statistical machine learning. Exact Bayesian calculations are often not feasible in practice, however. A number of approximate Bayesian methods have been proposed to make such calculations practical, among them the variational Bayesian (VB) approach. The VB approach, while useful, can nevertheless suffer from slow convergence to the approximate solution. To address this problem, we propose Parameter-eXpanded Variational Bayesian (PX-VB) methods to speed up VB. The new algorithm is inspired by parameter-expanded expectation maximization (PX-EM) and parameterexpanded data augmentation (PX-DA). Similar to PX-EM and -DA, PX-VB expands a model with auxiliary variables to reduce the coupling between variables in the original model. We analyze the convergence rates of VB and PX-VB and demonstrate the superior convergence rates of PX-VB in variational probit regression and automatic relevance determination.

This paper develops a multi-frame image super-resolution approach from a Bayesian viewpoint by marginalizing over the unknown registration parameters relating the set of input low-resolution views. In Tipping and Bishop's Bayesian image super-resolution approach [16], the marginalization was over the superresolution image, necessitating the use of an unfavorable image prior. By integrating over the registration parameters rather than the high-resolution image, our method allows for more realistic prior distributions, and also reduces the dimension of the integral considerably, removing the main computational bottleneck of the other algorithm. In addition to the motion model used by Tipping and Bishop, illumination components are introduced into the generative model, allowing us to handle changes in lighting as well as motion. We show results on real and synthetic datasets to illustrate the efficacy of this approach.

We develop and analyze game-theoretic algorithms for predicting coordinate binding of multiple DNA binding regulators. The allocation of proteins to local neighborhoods and to sites is carried out with resource constraints while explicating competing and coordinate binding relations among proteins with affinity to the site or region. The focus of this paper is on mathematical foundations of the approach. We also briefly demonstrate the approach in the context of the λ-phage switch.

Scoring structures of undirected graphical models by means of evaluating the marginal likelihood is very hard. The main reason is the presence of the partition function which is intractable to evaluate, let alone integrate over. We propose to approximate the marginal likelihood by employing two levels of approximation: we assume normality of the posterior (the Laplace approximation) and approximate all remaining intractable quantities using belief propagation and the linear response approximation.

We rigorously establish a close relationship between message passing algorithms and models of neurodynamics by showing that the equations of a continuous Hopfield network can be derived from the equations of belief propagation on a binary Markov random field. As Hopfield networks are equipped with a Lyapunov function, convergence is guaranteed. As a consequence, in the limit of many weak connections per neuron, Hopfield networks exactly implement a continuous-time variant of belief propagation starting from message initialisations that prevent from running into convergence problems. Our results lead to a better understanding of the role of message passing algorithms in real biological neural networks.

Blind source separation, i.e. the extraction of unknown sources from a set of given signals, is relevant for many applications. A special case of this problem is dimension reduction, where the goal is to approximate a given set of signals by superpositions of a minimal number of sources. Since in this case the signals outnumber the sources the problem is over-determined. Most popular approaches for addressing this problem are based on purely linear mixing models. However, many applications like the modeling of acoustic signals, EMG signals, or movement trajectories, require temporal shift-invariance of the extracted components. This case has only rarely been treated in the computational literature, and specifically for the case of dimension reduction almost no algorithms have been proposed. We present a new algorithm for the solution of this problem, which is based on a timefrequency transformation (Wigner-Ville distribution) of the generative model. We show that this algorithm outperforms classical source separation algorithms for linear mixtures, and also a related method for mixtures with delays. In addition, applying the new algorithm to trajectories of human gaits, we demonstrate that it is suitable for the extraction of spatiotemporal components that are easier to interpret than components extracted with other classical algorithms.

We use multi-electrode recordings from cat primary visual cortex and investigate whether a simple linear classifier can extract information about the presented stimuli. We find that information is extractable and that it even lasts for several hundred milliseconds after the stimulus has been removed. In a fast sequence of stimulus presentation, information about both new and old stimuli is present simultaneously and nonlinear relations between these stimuli can be extracted. These results suggest nonlinear properties of cortical representations. The important implications of these properties for the nonlinear brain theory are discussed.

The additive clustering model is widely used to infer the features of a set of stimuli from their similarities, on the assumption that similarity is a weighted linear function of common features. This paper develops a fully Bayesian formulation of the additive clustering model, using methods from nonparametric Bayesian statistics to allow the number of features to vary. We use this to explore several approaches to parameter estimation, showing that the nonparametric Bayesian approach provides a straightforward way to obtain estimates of both the number of features used in producing similarity judgments and their importance.

One of the intuitions underlying many graph-based methods for clustering and semi-supervised learning, is that class or cluster boundaries pass through areas of low probability density. In this paper we provide some formal analysis of that notion for a probability distribution. We introduce a notion of weighted boundary volume, which measures the length of the class/cluster boundary weighted by the density of the underlying probability distribution. We show that sizes of the cuts of certain commonly used data adjacency graphs converge to this continuous weighted volume of the boundary.

Spectral clustering methods are common graph-based approaches to clustering of data. Spectral clustering algorithms typically start from local information encoded in a weighted graph on the data and cluster according to the global eigenvectors of the corresponding (normalized) similarity matrix. One contribution of this paper is to present fundamental limitations of this general local to global approach. We show that based only on local information, the normalized cut functional is not a suitable measure for the quality of clustering. Further, even with a suitable similarity measure, we show that the first few eigenvectors of such adjacency matrices cannot successfully cluster datasets that contain structures at different scales of size and density. Based on these findings, a second contribution of this paper is a novel diffusion based measure to evaluate the coherence of individual clusters. Our measure can be used in conjunction with any bottom-up graph-based clustering method, it is scale-free and can determine coherent clusters at all scales. We present both synthetic examples and real image segmentation problems where various spectral clustering algorithms fail. In contrast, using this coherence measure finds the expected clusters at all scales.