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.

Power quality (PQ) analysis describes the non-pure electric signals that are usually present in electric power systems. The automatic recognition of PQ disturbances can be seen as a pattern recognition problem, in which different types of waveform distortion are differentiated based on their features. Similar to other quasi-stationary signals, PQ disturbances can be decomposed into time-frequency dependent components by using time-frequency or time-scale transforms, also known as dictionaries. These dictionaries are used in the feature extraction step in pattern recognition systems. Short-time Fourier, Wavelets and Stockwell transforms are some of the most common dictionaries used in the PQ community, aiming to achieve a better signal representation. To the best of our knowledge, previous works about PQ disturbance classification have been restricted to the use of one among several available dictionaries. Taking advantage of the theory behind sparse linear models (SLM), we introduce a sparse method for PQ representation, starting from overcomplete dictionaries. In particular, we apply Group Lasso. We employ different types of time-frequency (or time-scale) dictionaries to characterize the PQ disturbances, and evaluate their performance under different pattern recognition algorithms. We show that the SLM reduce the PQ classification complexity promoting sparse basis selection, and improving the classification accuracy.

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 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 the context of sparse recovery, it is known that most of existing regularizers such as $\ell_1$ suffer from some bias incurred by some leading entries (in magnitude) of the associated vector. To neutralize this bias, we propose a class of models with partial regularizers for recovering a sparse solution of a linear system. We show that every local minimizer of these models is sufficiently sparse or the magnitude of all its nonzero entries is above a uniform constant depending only on the data of the linear system. Moreover, for a class of partial regularizers, any global minimizer of these models is a sparsest solution to the linear system. We also establish some sufficient conditions for local or global recovery of the sparsest solution to the linear system, among which one of the conditions is weaker than the best known restricted isometry property (RIP) condition for sparse recovery by $\ell_1$. In addition, a first-order feasible augmented Lagrangian (FAL) method is proposed for solving these models, in which each subproblem is solved by a nonmonotone proximal gradient (NPG) method. Despite the complication of the partial regularizers, we show that each proximal subproblem in NPG can be solved as a certain number of one-dimensional optimization problems, which usually have a closed-form solution. We also show that any accumulation point of the sequence generated by FAL is a first-order stationary point of the models. Numerical results on compressed sensing and sparse logistic regression demonstrate that the proposed models substantially outperform the widely used ones in the literature in terms of solution quality.

Deep learning takes advantage of large datasets and computationally efficient training algorithms to outperform other approaches at various machine learning tasks. However, imperfections in the training phase of deep neural networks make them vulnerable to adversarial samples: inputs crafted by adversaries with the intent of causing deep neural networks to misclassify. In this work, we formalize the space of adversaries against deep neural networks (DNNs) and introduce a novel class of algorithms to craft adversarial samples based on a precise understanding of the mapping between inputs and outputs of DNNs. In an application to computer vision, we show that our algorithms can reliably produce samples correctly classified by human subjects but misclassified in specific targets by a DNN with a 97% adversarial success rate while only modifying on average 4.02% of the input features per sample. We then evaluate the vulnerability of different sample classes to adversarial perturbations by defining a hardness measure. Finally, we describe preliminary work outlining defenses against adversarial samples by defining a predictive measure of distance between a benign input and a target classification.

Large sample size brings the computation bottleneck for modern data analysis. Subsampling is one of efficient strategies to handle this problem. In previous studies, researchers make more fo- cus on subsampling with replacement (SSR) than on subsampling without replacement (SSWR). In this paper we investigate a kind of SSWR, poisson subsampling (PSS), for fast algorithm in ordinary least-square problem. We establish non-asymptotic property, i.e, the error bound of the correspond- ing subsample estimator, which provide a tradeoff between computation cost and approximation efficiency. Besides the non-asymptotic result, we provide asymptotic consistency and normality of the subsample estimator. Methodologically, we propose a two-step subsampling algorithm, which is efficient with respect to a statistical objective and independent on the linear model assumption.. Synthetic and real data are used to empirically study our proposed subsampling strategies. We argue by these empirical studies that, (1) our proposed two-step algorithm has obvious advantage when the assumed linear model does not accurate, and (2) the PSS strategy performs obviously better than SSR when the subsampling ratio increases.

We consider convex-concave saddle point problems with a separable structure and non-strongly convex functions. We propose an efficient stochastic block coordinate descent method using adaptive primal-dual updates, which enables flexible parallel optimization for large-scale problems. Our method shares the efficiency and flexibility of block coordinate descent methods with the simplicity of primal-dual methods and utilizing the structure of the separable convex-concave saddle point problem. It is capable of solving a wide range of machine learning applications, including robust principal component analysis, Lasso, and feature selection by group Lasso, etc. Theoretically and empirically, we demonstrate significantly better performance than state-of-the-art methods in all these applications.

We present a simple noise-robust margin-based active learning algorithm to find homogeneous (passing the origin) linear separators and analyze its error convergence when labels are corrupted by noise. We show that when the imposed noise satisfies the Tsybakov low noise condition (Mammen, Tsybakov, and others 1999; Tsybakov 2004) the algorithm is able to adapt to unknown level of noise and achieves optimal statistical rate up to poly-logarithmic factors. We also derive lower bounds for margin based active learning algorithms under Tsybakov noise conditions (TNC) for the membership query synthesis scenario (Angluin 1988). Our result implies lower bounds for the stream based selective sampling scenario (Cohn 1990) under TNC for some fairly simple data distributions. Quite surprisingly, we show that the sample complexity cannot be improved even if the underlying data distribution is as simple as the uniform distribution on the unit ball. Our proof involves the construction of a well separated hypothesis set on the d-dimensional unit ball along with carefully designed label distributions for the Tsybakov noise condition. Our analysis might provide insights for other forms of lower bounds as well.

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.