We consider robust covariance estimation with group symmetry constraints. Non-Gaussian covariance estimation, e.g., Tyler scatter estimator and Multivariate Generalized Gaussian distribution methods, usually involve non-convex minimization problems. Recently, it was shown that the underlying principle behind their success is an extended form of convexity over the geodesics in the manifold of positive definite matrices. A modern approach to improve estimation accuracy is to exploit prior knowledge via additional constraints, e.g., restricting the attention to specific classes of covariances which adhere to prior symmetry structures. In this paper, we prove that such group symmetry constraints are also geodesically convex and can therefore be incorporated into various non-Gaussian covariance estimators. Practical examples of such sets include: circulant, persymmetric and complex/quaternion proper structures. We provide a simple numerical technique for finding maximum likelihood estimates under such constraints, and demonstrate their performance advantage using synthetic experiments.

The L1-regularized Gaussian maximum likelihood estimator (MLE) has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix, or alternatively the underlying graph structure of a Gaussian Markov Random Field, from very limited samples. We propose a novel algorithm for solving the resulting optimization problem which is a regularized log-determinant program. In contrast to recent state-of-the-art methods that largely use first order gradient information, our algorithm is based on Newton's method and employs a quadratic approximation, but with some modifications that leverage the structure of the sparse Gaussian MLE problem. We show that our method is superlinearly convergent, and present experimental results using synthetic and real-world application data that demonstrate the considerable improvements in performance of our method when compared to other state-of-the-art methods.

The performance of a modulation classifier is highly sensitive to channel signal-to-noise ratio (SNR). In this paper, we focus on amplitude-phase modulations and propose a modulation classification framework based on centralized data fusion using multiple radios and the hybrid maximum likelihood (ML) approach. In order to alleviate the computational complexity associated with ML estimation, we adopt the Expectation Maximization (EM) algorithm. Due to SNR diversity, the proposed multi-radio framework provides robustness to channel SNR. Numerical results show the superiority of the proposed approach with respect to single radio approaches as well as to modulation classifiers using moments based estimators.

Latent Gaussian models (LGMs) are widely used in statistics and machine learning. Bayesian inference in non-conjugate LGMs is difficult due to intractable integrals involving the Gaussian prior and non-conjugate likelihoods. Algorithms based on variational Gaussian (VG) approximations are widely employed since they strike a favorable balance between accuracy, generality, speed, and ease of use. However, the structure of the optimization problems associated with these approximations remains poorly understood, and standard solvers take too long to converge. We derive a novel dual variational inference approach that exploits the convexity property of the VG approximations. We obtain an algorithm that solves a convex optimization problem, reduces the number of variational parameters, and converges much faster than previous methods. Using real-world data, we demonstrate these advantages on a variety of LGMs, including Gaussian process classification, and latent Gaussian Markov random fields.

We propose a technique for increasing the efficiency of gradient-based inference and learning in Bayesian networks with multiple layers of continuous latent vari- ables. We show that, in many cases, it is possible to express such models in an auxiliary form, where continuous latent variables are conditionally deterministic given their parents and a set of independent auxiliary variables. Variables of mod- els in this auxiliary form have much larger Markov blankets, leading to significant speedups in gradient-based inference, e.g. rapid mixing Hybrid Monte Carlo and efficient gradient-based optimization. The relative efficiency is confirmed in ex- periments.

The accurate prediction of time-changing covariances is an important problem in the modeling of multivariate financial data. However, some of the most popular models suffer from a) overfitting problems and multiple local optima, b) failure to capture shifts in market conditions and c) large computational costs. To address these problems we introduce a novel dynamic model for time-changing covariances. Over-fitting and local optima are avoided by following a Bayesian approach instead of computing point estimates. Changes in market conditions are captured by assuming a diffusion process in parameter values, and finally computationally efficient and scalable inference is performed using particle filters. Experiments with financial data show excellent performance of the proposed method with respect to current standard models.

Probabilistic programming allows specification of probabilistic models in a declarative manner. Recently, several new software systems and languages for probabilistic programming have been developed on the basis of newly developed and improved methods for approximate inference in probabilistic models. In this contribution a probabilistic model for an idealized dark matter localization problem is described. We first derive the probabilistic model for the inference of dark matter locations and masses, and then show how this model can be implemented using BUGS and Infer.NET, two software systems for probabilistic programming. Finally, the different capabilities of both systems are discussed. The presented dark matter model includes mainly non-conjugate factors, thus, it is difficult to implement this model with Infer.NET.

Major advances in Question Answering technology were needed for IBM Watson to play Jeopardy! at championship level -- the show requires rapid-fire answers to challenging natural language questions, broad general knowledge, high precision, and accurate confidence estimates. In addition, Jeopardy! features four types of decision making carrying great strategic importance: (1) Daily Double wagering; (2) Final Jeopardy wagering; (3) selecting the next square when in control of the board; (4) deciding whether to attempt to answer, i.e., "buzz in." Using sophisticated strategies for these decisions, that properly account for the game state and future event probabilities, can significantly boost a player's overall chances to win, when compared with simple "rule of thumb" strategies. This article presents our approach to developing Watson's game-playing strategies, comprising development of a faithful simulation model, and then using learning and Monte-Carlo methods within the simulator to optimize Watson's strategic decision-making. After giving a detailed description of each of our game-strategy algorithms, we then focus in particular on validating the accuracy of the simulator's predictions, and documenting performance improvements using our methods. Quantitative performance benefits are shown with respect to both simple heuristic strategies, and actual human contestant performance in historical episodes. We further extend our analysis of human play to derive a number of valuable and counterintuitive examples illustrating how human contestants may improve their performance on the show.

In data analysis, latent variables play a central role because they help provide powerful insights into a wide variety of phenomena, ranging from biological to human sciences. The latent tree model, a particular type of probabilistic graphical models, deserves attention. Its simple structure - a tree - allows simple and efficient inference, while its latent variables capture complex relationships. In the past decade, the latent tree model has been subject to significant theoretical and methodological developments. In this review, we propose a comprehensive study of this model. First we summarize key ideas underlying the model. Second we explain how it can be efficiently learned from data. Third we illustrate its use within three types of applications: latent structure discovery, multidimensional clustering, and probabilistic inference. Finally, we conclude and give promising directions for future researches in this field.

In this paper is proposed a new heuristic approach belonging to the field of evolutionary Estimation of Distribution Algorithms (EDAs). EDAs builds a probability model and a set of solutions is sampled from the model which characterizes the distribution of such solutions. The main framework of the proposed method is an estimation of distribution algorithm, in which an adaptive Gibbs sampling is used to generate new promising solutions and, in combination with a local search strategy, it improves the individual solutions produced in each iteration. The Estimation of Distribution Algorithm with Adaptive Gibbs Sampling we are proposing in this paper is called AGEDA. We experimentally evaluate and compare this algorithm against two deterministic procedures and several stochastic methods in three well known test problems for unconstrained global optimization. It is empirically shown that our heuristic is robust in problems that involve three central aspects that mainly determine the difficulty of global optimization problems, namely high-dimensionality, multi-modality and non-smoothness.