We present a probabilistic variant of the recently introduced maxout unit. The success of deep neural networks utilizing maxout can partly be attributed to favorable performance under dropout, when compared to rectified linear units. It however also depends on the fact that each maxout unit performs a pooling operation over a group of linear transformations and is thus partially invariant to changes in its input. Starting from this observation we ask the question: Can the desirable properties of maxout units be preserved while improving their invariance properties ? We argue that our probabilistic maxout (probout) units successfully achieve this balance. We quantitatively verify this claim and report classification performance matching or exceeding the current state of the art on three challenging image classification benchmarks (CIFAR-10, CIFAR-100 and SVHN).

The episodic, irregular and asynchronous nature of medical data render them difficult substrates for standard machine learning algorithms. We would like to abstract away this difficulty for the class of time-stamped categorical variables (or events) by modeling them as a renewal process and inferring a probability density over continuous, longitudinal, nonparametric intensity functions modulating that process. Several methods exist for inferring such a density over intensity functions, but either their constraints and assumptions prevent their use with our potentially bursty event streams, or their time complexity renders their use intractable on our long-duration observations of high-resolution events, or both. In this paper we present a new and efficient method for inferring a distribution over intensity functions that uses direct numeric integration and smooth interpolation over Gaussian processes. We demonstrate that our direct method is up to twice as accurate and two orders of magnitude more efficient than the best existing method (thinning). Importantly, the direct method can infer intensity functions over the full range of bursty to memoryless to regular events, which thinning and many other methods cannot. Finally, we apply the method to clinical event data and demonstrate the face-validity of the abstraction, which is now amenable to standard learning algorithms.

This paper considers the problem of low-dimensional visualisation of very high dimensional information sources for the purpose of situation awareness in the maritime environment. In response to the requirement for human decision support aids to reduce information overload (and specifically, data amenable to inter-point relative similarity measures) appropriate to the below-water maritime domain, we are investigating a preliminary prototype topographic visualisation model. The focus of the current paper is on the mathematical problem of exploiting a relative dissimilarity representation of signals in a visual informatics mapping model, driven by real-world sonar systems. An independent source model is used to analyse the sonar beams from which a simple probabilistic input model to represent uncertainty is mapped to a latent visualisation space where data uncertainty can be accommodated. The use of euclidean and non-euclidean measures are used and the motivation for future use of non-euclidean measures is made. Concepts are illustrated using a simulated 64 beam weak SNR dataset with realistic sonar targets.

Statistical and machine-learning algorithms are frequently applied to high-dimensional data. In many of these applications data is scarce, and often much more costly than computation time. We provide the first sample-efficient polynomial-time estimator for high-dimensional spherical Gaussian mixtures. For mixtures of any $k$ $d$-dimensional spherical Gaussians, we derive an intuitive spectral-estimator that uses $\mathcal{O}_k\bigl(\frac{d\log^2d}{\epsilon^4}\bigr)$ samples and runs in time $\mathcal{O}_{k,\epsilon}(d^3\log^5 d)$, both significantly lower than previously known. The constant factor $\mathcal{O}_k$ is polynomial for sample complexity and is exponential for the time complexity, again much smaller than what was previously known. We also show that $\Omega_k\bigl(\frac{d}{\epsilon^2}\bigr)$ samples are needed for any algorithm. Hence the sample complexity is near-optimal in the number of dimensions. We also derive a simple estimator for one-dimensional mixtures that uses $\mathcal{O}\bigl(\frac{k \log \frac{k}{\epsilon} }{\epsilon^2} \bigr)$ samples and runs in time $\widetilde{\mathcal{O}}\left(\bigl(\frac{k}{\epsilon}\bigr)^{3k+1}\right)$. Our other technical contributions include a faster algorithm for choosing a density estimate from a set of distributions, that minimizes the $\ell_1$ distance to an unknown underlying distribution.

Hierarchical statistical models are widely employed in information science and data engineering. The models consist of two types of variables: observable variables that represent the given data and latent variables for the unobservable labels. An asymptotic analysis of the models plays an important role in evaluating the learning process; the result of the analysis is applied not only to theoretical but also to practical situations, such as optimal model selection and active learning. There are many studies of generalization errors, which measure the prediction accuracy of the observable variables. However, the accuracy of estimating the latent variables has not yet been elucidated. For a quantitative evaluation of this, the present paper formulates distribution-based functions for the errors in the estimation of the latent variables. The asymptotic behavior is analyzed for both the maximum likelihood and the Bayes methods.

This paper presents an implementation of the Imperialist Competitive Algorithm (ICA) for solving the fuzzy random portfolio selection problem where the asset returns are represented by fuzzy random variables. Portfolio Optimization is an important research field in modern finance. By using the necessity-based model, fuzzy random variables reformulate to the linear programming and ICA will be designed to find the optimum solution. To show the efficiency of the proposed method, a numerical example illustrates the whole idea on implementation of ICA for fuzzy random portfolio selection problem.

In this paper, we provide more evidence for the contention that logical consequence should be understood in normative terms. Hartry Field and John MacFarlane covered the classical case. We extend their work, examining what it means for an agent to be obliged to infer a conclusion when faced with uncertain information or reasoning within a non-monotonic, defeasible, logical framework (which allows e. g. for inference to be drawn from premises considered true unless evidence to the contrary is presented).

We investigate the Student-t process as an alternative to the Gaussian process as a nonparametric prior over functions. We derive closed form expressions for the marginal likelihood and predictive distribution of a Student-t process, by integrating away an inverse Wishart process prior over the covariance kernel of a Gaussian process model. We show surprising equivalences between different hierarchical Gaussian process models leading to Student-t processes, and derive a new sampling scheme for the inverse Wishart process, which helps elucidate these equivalences. Overall, we show that a Student-t process can retain the attractive properties of a Gaussian process -- a nonparametric representation, analytic marginal and predictive distributions, and easy model selection through covariance kernels -- but has enhanced flexibility, and predictive covariances that, unlike a Gaussian process, explicitly depend on the values of training observations. We verify empirically that a Student-t process is especially useful in situations where there are changes in covariance structure, or in applications like Bayesian optimization, where accurate predictive covariances are critical for good performance. These advantages come at no additional computational cost over Gaussian processes.

Recent work introduced the cube-and-conquer technique to solve hard SAT instances. It partitions the search space into cubes using a lookahead solver. Each cube is tackled by a conflict-driven clause learning (CDCL) solver. Crucial for strong performance is the cutoff heuristic that decides when to switch from lookahead to CDCL. Yet, this offline heuristic is far from ideal. In this paper, we present a novel hybrid solver that applies the cube and conquer steps simultaneously. A lookahead and a CDCL solver work together on each cube, while communication is restricted to synchronization. Our concurrent cube-and-conquer solver can solve many instances faster than pure lookahead, pure CDCL and offline cube-and-conquer, and can abort early in favor of a pure CDCL search if an instance is not suitable for cube-and-conquer techniques.

The split Bregman (SB) method [T. Goldstein and S. Osher, SIAM J. Imaging Sci., 2 (2009), pp. 323-43] is a fast splitting-based algorithm that solves image reconstruction problems with general l1, e.g., total-variation (TV) and compressed sensing (CS), regularizations by introducing a single variable split to decouple the data-fitting term and the regularization term, yielding simple subproblems that are separable (or partially separable) and easy to minimize. Several convergence proofs have been proposed, and these proofs either impose a "full column rank" assumption to the split or assume exact updates in all subproblems. However, these assumptions are impractical in many applications such as the X-ray computed tomography (CT) image reconstructions, where the inner least-squares problem usually cannot be solved efficiently due to the highly shift-variant Hessian. In this paper, we show that when the data-fitting term is quadratic, the SB method is a convergent alternating direction method of multipliers (ADMM), and a straightforward convergence proof with inexact updates is given using [J. Eckstein and D. P. Bertsekas, Mathematical Programming, 55 (1992), pp. 293-318, Theorem 8]. Furthermore, since the SB method is just a special case of an ADMM algorithm, it seems likely that the ADMM algorithm will be faster than the SB method if the augmented Largangian (AL) penalty parameters are selected appropriately. To have a concrete example, we conduct a convergence rate analysis of the ADMM algorithm using two splits for image restoration problems with quadratic data-fitting term and regularization term. According to our analysis, we can show that the two-split ADMM algorithm can be faster than the SB method if the AL penalty parameter of the SB method is suboptimal. Numerical experiments were conducted to verify our analysis.