In this paper we discuss Bayesian nonconvex penalization for sparse learning problems. We explore a nonparametric formulation for latent shrinkage parameters using subordinators which are one-dimensional L\'{e}vy processes. We particularly study a family of continuous compound Poisson subordinators and a family of discrete compound Poisson subordinators. We exemplify four specific subordinators: Gamma, Poisson, negative binomial and squared Bessel subordinators. The Laplace exponents of the subordinators are Bernstein functions, so they can be used as sparsity-inducing nonconvex penalty functions. We exploit these subordinators in regression problems, yielding a hierarchical model with multiple regularization parameters. We devise ECME (Expectation/Conditional Maximization Either) algorithms to simultaneously estimate regression coefficients and regularization parameters. The empirical evaluation of simulated data shows that our approach is feasible and effective in high-dimensional data analysis.

Adaptive optimal control using value iteration (VI) initiated from a stabilizing policy is theoretically analyzed in various aspects including the continuity of the result, the stability of the system operated using any single/constant resulting control policy, the stability of the system operated using the evolving/time-varying control policy, the convergence of the algorithm, and the optimality of the limit function. Afterwards, the effect of presence of approximation errors in the involved function approximation processes is incorporated and another set of results for boundedness of the approximate VI as well as stability of the system operated under the results for both cases of applying a single policy or an evolving policy are derived. A feature of the presented results is providing estimations of the region of attraction so that if the initial condition is within the region, the whole trajectory will remain inside it and hence, the function approximation results will be reliable.

While looking for abductive explanations of a given set of manifestations, an ordering between possible solutions is often assumed. The complexity of finding/verifying optimal solutions is already known. In this paper we consider the computational complexity of finding second-best solutions. We consider different orderings, and consider also different possible definitions of what a second-best solution is.

We provide scientific foundations for athletic performance prediction on an individual level, exposing the phenomenology of individual athletic running performance in the form of a low-rank model dominated by an individual power law. We present, evaluate, and compare a selection of methods for prediction of individual running performance, including our own, \emph{local matrix completion} (LMC), which we show to perform best. We also show that many documented phenomena in quantitative sports science, such as the form of scoring tables, the success of existing prediction methods including Riegel's formula, the Purdy points scheme, the power law for world records performances and the broken power law for world record speeds may be explained on the basis of our findings in a unified way.

We describe a method for removing the effect of confounders in order to reconstruct a latent quantity of interest. The method, referred to as half-sibling regression, is inspired by recent work in causal inference using additive noise models. We provide a theoretical justification and illustrate the potential of the method in a challenging astronomy application.

Markov chain Monte Carlo methods are often deemed too computationally intensive to be of any practical use for big data applications, and in particular for inference on datasets containing a large number $n$ of individual data points, also known as tall datasets. In scenarios where data are assumed independent, various approaches to scale up the Metropolis-Hastings algorithm in a Bayesian inference context have been recently proposed in machine learning and computational statistics. These approaches can be grouped into two categories: divide-and-conquer approaches and, subsampling-based algorithms. The aims of this article are as follows. First, we present a comprehensive review of the existing literature, commenting on the underlying assumptions and theoretical guarantees of each method. Second, by leveraging our understanding of these limitations, we propose an original subsampling-based approach which samples from a distribution provably close to the posterior distribution of interest, yet can require less than $O(n)$ data point likelihood evaluations at each iteration for certain statistical models in favourable scenarios. Finally, we have only been able so far to propose subsampling-based methods which display good performance in scenarios where the Bernstein-von Mises approximation of the target posterior distribution is excellent. It remains an open challenge to develop such methods in scenarios where the Bernstein-von Mises approximation is poor.

The Gaussian process latent variable model (GP-LVM) is a popular approach to non-linear probabilistic dimensionality reduction. One design choice for the model is the number of latent variables. We present a spike and slab prior for the GP-LVM and propose an efficient variational inference procedure that gives a lower bound of the log marginal likelihood. The new model provides a more principled approach for selecting latent dimensions than the standard way of thresholding the length-scale parameters. The effectiveness of our approach is demonstrated through experiments on real and simulated data. Further, we extend multi-view Gaussian processes that rely on sharing latent dimensions (known as manifold relevance determination) with spike and slab priors. This allows a more principled approach for selecting a subset of the latent space for each view of data. The extended model outperforms the previous state-of-the-art when applied to a cross-modal multimedia retrieval task.

Improvement guarantees for semi-supervised classifiers can currently only be given under restrictive conditions on the data. We propose a general way to perform semi-supervised parameter estimation for likelihood-based classifiers for which, on the full training set, the estimates are never worse than the supervised solution in terms of the log-likelihood. We argue, moreover, that we may expect these solutions to really improve upon the supervised classifier in particular cases. In a worked-out example for LDA, we take it one step further and essentially prove that its semi-supervised version is strictly better than its supervised counterpart. The two new concepts that form the core of our estimation principle are contrast and pessimism. The former refers to the fact that our objective function takes the supervised estimates into account, enabling the semi-supervised solution to explicitly control the potential improvements over this estimate. The latter refers to the fact that our estimates are conservative and therefore resilient to whatever form the true labeling of the unlabeled data takes on. Experiments demonstrate the improvements in terms of both the log-likelihood and the classification error rate on independent test sets.

In this paper we propose the first non-parametric Bayesian model using Gaussian Processes to make inference on Poisson Point Processes without resorting to gridding the domain or to introducing latent thinning points. Unlike competing models that scale cubically and have a squared memory requirement in the number of data points, our model has a linear complexity and memory requirement. We propose an MCMC sampler and show that our model is faster, more accurate and generates less correlated samples than competing models on both synthetic and real-life data. Finally, we show that our model easily handles data sizes not considered thus far by alternate approaches.

Given a matrix $M$ (not necessarily nonnegative) and a factorization rank $r$, semi-nonnegative matrix factorization (semi-NMF) looks for a matrix $U$ with $r$ columns and a nonnegative matrix $V$ with $r$ rows such that $UV$ is the best possible approximation of $M$ according to some metric. In this paper, we study the properties of semi-NMF from which we develop exact and heuristic algorithms. Our contribution is threefold. First, we prove that the error of a semi-NMF of rank $r$ has to be smaller than the best unconstrained approximation of rank $r-1$. This leads us to a new initialization procedure based on the singular value decomposition (SVD) with a guarantee on the quality of the approximation. Second, we propose an exact algorithm (that is, an algorithm that finds an optimal solution), also based on the SVD, for a certain class of matrices (including nonnegative irreducible matrices) from which we derive an initialization for matrices not belonging to that class. Numerical experiments illustrate that this second approach performs extremely well, and allows us to compute optimal semi-NMF decompositions in many situations. Finally, we analyze the computational complexity of semi-NMF proving its NP-hardness, already in the rank-one case (that is, for $r = 1$), and we show that semi-NMF is sometimes ill-posed (that is, an optimal solution does not exist).