Latent variable time-series models are among the most heavily used tools from machine learning and applied statistics. These models have the advantage of learning latent structure both from noisy observations and from the temporal ordering in the data, where it is assumed that meaningful correlation structure exists across time. A few highly-structured models, such as the linear dynamical system with linear-Gaussian observations, have closed-form inference procedures (e.g. the Kalman Filter), but this case is an exception to the general rule that exact posterior inference in more complex generative models is intractable. Consequently, much work in time-series modeling focuses on approximate inference procedures for one particular class of models. Here, we extend recent developments in stochastic variational inference to develop a `black-box' approximate inference technique for latent variable models with latent dynamical structure. We propose a structured Gaussian variational approximate posterior that carries the same intuition as the standard Kalman filter-smoother but, importantly, permits us to use the same inference approach to approximate the posterior of much more general, nonlinear latent variable generative models. We show that our approach recovers accurate estimates in the case of basic models with closed-form posteriors, and more interestingly performs well in comparison to variational approaches that were designed in a bespoke fashion for specific non-conjugate models.

We develop parallel predictive entropy search (PPES), a novel algorithm for Bayesian optimization of expensive black-box objective functions. At each iteration, PPES aims to select a batch of points which will maximize the information gain about the global maximizer of the objective. Well known strategies exist for suggesting a single evaluation point based on previous observations, while far fewer are known for selecting batches of points to evaluate in parallel. The few batch selection schemes that have been studied all resort to greedy methods to compute an optimal batch. To the best of our knowledge, PPES is the first non-greedy batch Bayesian optimization strategy. We demonstrate the benefit of this approach in optimization performance on both synthetic and real world applications, including problems in machine learning, rocket science and robotics.

In this paper we review the concepts of Bayesian evidence and Bayes factors, also known as log odds ratios, and their application to model selection. The theory is presented along with a discussion of analytic, approximate and numerical techniques. Specific attention is paid to the Laplace approximation, variational Bayes, importance sampling, thermodynamic integration, and nested sampling and its recent variants. Analogies to statistical physics, from which many of these techniques originate, are discussed in order to provide readers with deeper insights that may lead to new techniques. The utility of Bayesian model testing in the domain sciences is demonstrated by presenting four specific practical examples considered within the context of signal processing in the areas of signal detection, sensor characterization, scientific model selection and molecular force characterization.

We introduce a framework for analyzing transductive combination of Gaussian process (GP) experts, where independently trained GP experts are combined in a way that depends on test point location, in order to scale GPs to big data. The framework provides some theoretical justification for the generalized product of GP experts (gPoE-GP) which was previously shown to work well in practice but lacks theoretical basis. Based on the proposed framework, an improvement over gPoE-GP is introduced and empirically validated.

We give an algorithm for completing an order-$m$ symmetric low-rank tensor from its multilinear entries in time roughly proportional to the number of tensor entries. We apply our tensor completion algorithm to the problem of learning mixtures of product distributions over the hypercube, obtaining new algorithmic results. If the centers of the product distribution are linearly independent, then we recover distributions with as many as $\Omega(n)$ centers in polynomial time and sample complexity. In the general case, we recover distributions with as many as $\tilde\Omega(n)$ centers in quasi-polynomial time, answering an open problem of Feldman et al. (SIAM J. Comp.) for the special case of distributions with incoherent bias vectors. Our main algorithmic tool is the iterated application of a low-rank matrix completion algorithm for matrices with adversarially missing entries.

In this work, we propose a generalized product of experts (gPoE) framework for combining the predictions of multiple probabilistic models. We identify four desirable properties that are important for scalability, expressiveness and robustness, when learning and inferring with a combination of multiple models. Through analysis and experiments, we show that gPoE of Gaussian processes (GP) have these qualities, while no other existing combination schemes satisfy all of them at the same time. The resulting GP-gPoE is highly scalable as individual GP experts can be independently learned in parallel; very expressive as the way experts are combined depends on the input rather than fixed; the combined prediction is still a valid probabilistic model with natural interpretation; and finally robust to unreliable predictions from individual experts.

Despite its nonconvex nature, $\ell_0$ sparse approximation is desirable in many theoretical and application cases. We study the $\ell_0$ sparse approximation problem with the tool of deep learning, by proposing Deep $\ell_0$ Encoders. Two typical forms, the $\ell_0$ regularized problem and the $M$-sparse problem, are investigated. Based on solid iterative algorithms, we model them as feed-forward neural networks, through introducing novel neurons and pooling functions. Enforcing such structural priors acts as an effective network regularization. The deep encoders also enjoy faster inference, larger learning capacity, and better scalability compared to conventional sparse coding solutions. Furthermore, under task-driven losses, the models can be conveniently optimized from end to end. Numerical results demonstrate the impressive performances of the proposed encoders.

We propose a likelihood ratio based inferential framework for high dimensional semiparametric generalized linear models. This framework addresses a variety of challenging problems in high dimensional data analysis, including incomplete data, selection bias, and heterogeneous multitask learning. Our work has three main contributions. (i) We develop a regularized statistical chromatography approach to infer the parameter of interest under the proposed semiparametric generalized linear model without the need of estimating the unknown base measure function. (ii) We propose a new framework to construct post-regularization confidence regions and tests for the low dimensional components of high dimensional parameters. Unlike existing post-regularization inferential methods, our approach is based on a novel directional likelihood. In particular, the framework naturally handles generic regularized estimators with nonconvex penalty functions and it can be used to infer least false parameters under misspecified models. (iii) We develop new concentration inequalities and normal approximation results for U-statistics with unbounded kernels, which are of independent interest. We demonstrate the consequences of the general theory by using an example of missing data problem. Extensive simulation studies and real data analysis are provided to illustrate our proposed approach.

This paper presents a novel nonmyopic adaptive Gaussian process planning (GPP) framework endowed with a general class of Lipschitz continuous reward functions that can unify some active learning/sensing and Bayesian optimization criteria and offer practitioners some flexibility to specify their desired choices for defining new tasks/problems. In particular, it utilizes a principled Bayesian sequential decision problem framework for jointly and naturally optimizing the exploration-exploitation trade-off. In general, the resulting induced GPP policy cannot be derived exactly due to an uncountable set of candidate observations. A key contribution of our work here thus lies in exploiting the Lipschitz continuity of the reward functions to solve for a nonmyopic adaptive epsilon-optimal GPP (epsilon-GPP) policy. To plan in real time, we further propose an asymptotically optimal, branch-and-bound anytime variant of epsilon-GPP with performance guarantee. We empirically demonstrate the effectiveness of our epsilon-GPP policy and its anytime variant in Bayesian optimization and an energy harvesting task.

Class ambiguity is typical in image classification problems with a large number of classes. When classes are difficult to discriminate, it makes sense to allow k guesses and evaluate classifiers based on the top-k error instead of the standard zero-one loss. We propose top-k multiclass SVM as a direct method to optimize for top-k performance. Our generalization of the well-known multiclass SVM is based on a tight convex upper bound of the top-k error. We propose a fast optimization scheme based on an efficient projection onto the top-k simplex, which is of its own interest. Experiments on five datasets show consistent improvements in top-k accuracy compared to various baselines.