Recovery of low-rank matrices has recently seen significant activity in many areas of science and engineering, motivated by recent theoretical results for exact reconstruction guarantees and interesting practical applications. A number of methods have been developed for this recovery problem. However, a principled method for choosing the unknown target rank is generally not provided. In this paper, we present novel recovery algorithms for estimating low-rank matrices in matrix completion and robust principal component analysis based on sparse Bayesian learning (SBL) principles. Starting from a matrix factorization formulation and enforcing the low-rank constraint in the estimates as a sparsity constraint, we develop an approach that is very effective in determining the correct rank while providing high recovery performance. We provide connections with existing methods in other similar problems and empirical results and comparisons with current state-of-the-art methods that illustrate the effectiveness of this approach.

We present a new algorithm for exactly solving decision making problems represented as influence diagrams. We do not require the usual assumptions of no forgetting and regularity; this allows us to solve problems with simultaneous decisions and limited information. The algorithm is empirically shown to outperform a state-of-the-art algorithm on randomly generated problems of up to 150 variables and $10^{64}$ solutions. We show that the problem is NP-hard even if the underlying graph structure of the problem has small treewidth and the variables take on a bounded number of states, but that a fully polynomial time approximation scheme exists for these cases. Moreover, we show that the bound on the number of states is a necessary condition for any efficient approximation scheme.

We study linear models under heavy-tailed priors from a probabilistic viewpoint. Instead of computing a single sparse most probable (MAP) solution as in standard deterministic approaches, the focus in the Bayesian compressed sensing framework shifts towards capturing the full posterior distribution on the latent variables, which allows quantifying the estimation uncertainty and learning model parameters using maximum likelihood. The exact posterior distribution under the sparse linear model is intractable and we concentrate on variational Bayesian techniques to approximate it. Repeatedly computing Gaussian variances turns out to be a key requisite and constitutes the main computational bottleneck in applying variational techniques in large-scale problems. We leverage on the recently proposed Perturb-and-MAP algorithm for drawing exact samples from Gaussian Markov random fields (GMRF). The main technical contribution of our paper is to show that estimating Gaussian variances using a relatively small number of such efficiently drawn random samples is much more effective than alternative general-purpose variance estimation techniques. By reducing the problem of variance estimation to standard optimization primitives, the resulting variational algorithms are fully scalable and parallelizable, allowing Bayesian computations in extremely large-scale problems with the same memory and time complexity requirements as conventional point estimation techniques. We illustrate these ideas with experiments in image deblurring.

X, where f(x) is the gradient of f at x. This is called the convex regression problem. Convex regression can easily be modified to allow concave regression by multiplying all of the values by negative one. Convex regression problems are common in economics, operations research and reinforcement learning. In economics, production functions (Skiba 1978) and consumer preferences (Meyer & Pratt 1968) are often convex, while in operations research and reinforcement learning, value functions for stochastic optimization problems can be convex (Shapiro et al. 2009). If a problem is known to be convex, a convex regression estimate provides advantages over an unrestricted estimate. First, convexity is a powerful regularizer: it places strong conditions on the derivatives--and hence smoothness--of a function. Convexity constraints can substantially reduce overfitting and lead to more accurate predictions. Second, maintaining convexity allows the use of convex optimization solvers when the regression estimate is used in an objective function of an optimization problem. 1 Multivariate convex regression has received relatively little attention in the literature.

Robots operating in domestic environments generally need to interact with articulated objects, such as doors, cabinets, dishwashers or fridges. In this work, we present a novel, probabilistic framework for modeling articulated objects as kinematic graphs. Vertices in this graph correspond to object parts, while edges between them model their kinematic relationship. In particular, we present a set of parametric and non-parametric edge models and how they can robustly be estimated from noisy pose observations. We furthermore describe how to estimate the kinematic structure and how to use the learned kinematic models for pose prediction and for robotic manipulation tasks. We finally present how the learned models can be generalized to new and previously unseen objects. In various experiments using real robots with different camera systems as well as in simulation, we show that our approach is valid, accurate and efficient. Further, we demonstrate that our approach has a broad set of applications, in particular for the emerging fields of mobile manipulation and service robotics.

This article presents a survey of work on lifted graphical models. We review a general form for a lifted graphical model, a par-factor graph, and show how a number of existing statistical relational representations map to this formalism. We discuss inference algorithms, including lifted inference algorithms, that efficiently compute the answers to probabilistic queries. We also review work in learning lifted graphical models from data. It is our belief that the need for statistical relational models (whether it goes by that name or another) will grow in the coming decades, as we are inundated with data which is a mix of structured and unstructured, with entities and relations extracted in a noisy manner from text, and with the need to reason effectively with this data. We hope that this synthesis of ideas from many different research groups will provide an accessible starting point for new researchers in this expanding field.

In this work we introduce a mixture of GPs to address the data association problem, i.e. to label a group of observations according to the sources that generated them. Unlike several previously proposed GP mixtures, the novel mixture has the distinct characteristic of using no gating function to determine the association of samples and mixture components. Instead, all the GPs in the mixture are global and samples are clustered following "trajectories" across input space. We use a nonstandard variational Bayesian algorithm to efficiently recover sample labels and learn the hyperparameters. We show how multi-object tracking problems can be disambiguated and also explore the characteristics of the model in traditional regression settings. Keywords: Gaussian Processes, Marginalized Variational Inference, Bayesian Models 1. Introduction The data association problem arises in multi-target tracking scenarios.

We address the sparse signal recovery problem in the context of multiple measurement vectors (MMV) when elements in each nonzero row of the solution matrix are temporally correlated. Existing algorithms do not consider such temporal correlations and thus their performance degrades significantly with the correlations. In this work, we propose a block sparse Bayesian learning framework which models the temporal correlations. In this framework we derive two sparse Bayesian learning (SBL) algorithms, which have superior recovery performance compared to existing algorithms, especially in the presence of high temporal correlations. Furthermore, our algorithms are better at handling highly underdetermined problems and require less row-sparsity on the solution matrix. We also provide analysis of the global and local minima of their cost function, and show that the SBL cost function has the very desirable property that the global minimum is at the sparsest solution to the MMV problem. Extensive experiments also provide some interesting results that motivate future theoretical research on the MMV model.

We consider the problem of speaker diarization, the problem of segmenting an audio recording of a meeting into temporal segments corresponding to individual speakers. The problem is rendered particularly difficult by the fact that we are not allowed to assume knowledge of the number of people participating in the meeting. To address this problem, we take a Bayesian nonparametric approach to speaker diarization that builds on the hierarchical Dirichlet process hidden Markov model (HDP-HMM) of Teh et al. [J. Amer. Statist. Assoc. 101 (2006) 1566--1581]. Although the basic HDP-HMM tends to over-segment the audio data---creating redundant states and rapidly switching among them---we describe an augmented HDP-HMM that provides effective control over the switching rate. We also show that this augmentation makes it possible to treat emission distributions nonparametrically. To scale the resulting architecture to realistic diarization problems, we develop a sampling algorithm that employs a truncated approximation of the Dirichlet process to jointly resample the full state sequence, greatly improving mixing rates. Working with a benchmark NIST data set, we show that our Bayesian nonparametric architecture yields state-of-the-art speaker diarization results.

Inference in popular nonparametric Bayesian models typically relies on sampling or other approximations. This paper presents a general methodology for constructing novel tractable nonparametric Bayesian methods by applying the kernel trick to inference in a parametric Bayesian model. For example, Gaussian process regression can be derived this way from Bayesian linear regression. Despite the success of the Gaussian process framework, the kernel trick is rarely explicitly considered in the Bayesian literature. In this paper, we aim to fill this gap and demonstrate the potential of applying the kernel trick to tractable Bayesian parametric models in a wider context than just regression. As an example, we present an intuitive Bayesian kernel machine for density estimation that is obtained by applying the kernel trick to a Gaussian generative model in feature space.