Max-convolution is an important problem closely resembling standard convolution; as such, max-convolution occurs frequently across many fields. Here we extend the method with fastest known worst-case runtime, which can be applied to nonnegative vectors by numerically approximating the Chebyshev norm $\| \cdot \|_\infty$, and use this approach to derive two numerically stable methods based on the idea of computing $p$-norms via fast convolution: The first method proposed, with runtime in $O( k \log(k) \log(\log(k)) )$ (which is less than $18 k \log(k)$ for any vectors that can be practically realized), uses the $p$-norm as a direct approximation of the Chebyshev norm. The second approach proposed, with runtime in $O( k \log(k) )$ (although in practice both perform similarly), uses a novel null space projection method, which extracts information from a sequence of $p$-norms to estimate the maximum value in the vector (this is equivalent to querying a small number of moments from a distribution of bounded support in order to estimate the maximum). The $p$-norm approaches are compared to one another and are shown to compute an approximation of the Viterbi path in a hidden Markov model where the transition matrix is a Toeplitz matrix; the runtime of approximating the Viterbi path is thus reduced from $O( n k^2 )$ steps to $O( n $k \log(k))$ steps in practice, and is demonstrated by inferring the U.S. unemployment rate from the S&P 500 stock index.

Acyclic digraphs are the underlying representation of Bayesian networks, a widely used class of probabilistic graphical models. Learning the underlying graph from data is a way of gaining insights about the structural properties of a domain. Structure learning forms one of the inference challenges of statistical graphical models. MCMC methods, notably structure MCMC, to sample graphs from the posterior distribution given the data are probably the only viable option for Bayesian model averaging. Score modularity and restrictions on the number of parents of each node allow the graphs to be grouped into larger collections, which can be scored as a whole to improve the chain's convergence. Current examples of algorithms taking advantage of grouping are the biased order MCMC, which acts on the alternative space of permuted triangular matrices, and non ergodic edge reversal moves. Here we propose a novel algorithm, which employs the underlying combinatorial structure of DAGs to define a new grouping. As a result convergence is improved compared to structure MCMC, while still retaining the property of producing an unbiased sample. Finally the method can be combined with edge reversal moves to improve the sampler further.

As part of daily monitoring of human activities, wearable sensors and devices are becoming increasingly popular sources of data. With the advent of smartphones equipped with acceloremeter, gyroscope and camera; it is now possible to develop activity classification platforms everyone can use conveniently. In this paper, we propose a fast inference method for an unsupervised non-parametric time series model namely variational inference for sticky HDP-SLDS(Hierarchical Dirichlet Process Switching Linear Dynamical System). We show that the proposed algorithm can differentiate various indoor activities such as sitting, walking, turning, going up/down the stairs and taking the elevator using only the acceloremeter of an Android smartphone Samsung Galaxy S4. We used the front camera of the smartphone to annotate activity types precisely. We compared the proposed method with Hidden Markov Models with Gaussian emission probabilities on a dataset of 10 subjects. We showed that the efficacy of the stickiness property. We further compared the variational inference to the Gibbs sampler on the same model and show that variational inference is faster in one order of magnitude.

The use of mutual information as a similarity measure in agglomerative hierarchical clustering (AHC) raises an important issue: some correction needs to be applied for the dimensionality of variables. In this work, we formulate the decision of merging dependent multivariate normal variables in an AHC procedure as a Bayesian model comparison. We found that the Bayesian formulation naturally shrinks the empirical covariance matrix towards a matrix set a priori (e.g., the identity), provides an automated stopping rule, and corrects for dimensionality using a term that scales up the measure as a function of the dimensionality of the variables. Also, the resulting log Bayes factor is asymptotically proportional to the plug-in estimate of mutual information, with an additive correction for dimensionality in agreement with the Bayesian information criterion. We investigated the behavior of these Bayesian alternatives (in exact and asymptotic forms) to mutual information on simulated and real data. An encouraging result was first derived on simulations: the hierarchical clustering based on the log Bayes factor outperformed off-the-shelf clustering techniques as well as raw and normalized mutual information in terms of classification accuracy. On a toy example, we found that the Bayesian approaches led to results that were similar to those of mutual information clustering techniques, with the advantage of an automated thresholding. On real functional magnetic resonance imaging (fMRI) datasets measuring brain activity, it identified clusters consistent with the established outcome of standard procedures. On this application, normalized mutual information had a highly atypical behavior, in the sense that it systematically favored very large clusters. These initial experiments suggest that the proposed Bayesian alternatives to mutual information are a useful new tool for hierarchical clustering.

Measuring the impact of scientific articles is important for evaluating the research output of individual scientists, academic institutions and journals. While citations are raw data for constructing impact measures, there exist biases and potential issues if factors affecting citation patterns are not properly accounted for. In this work, we address the problem of field variation and introduce an article level metric useful for evaluating individual articles' visibility. This measure derives from joint probabilistic modeling of the content in the articles and the citations amongst them using latent Dirichlet allocation (LDA) and the mixed membership stochastic blockmodel (MMSB). Our proposed model provides a visibility metric for individual articles adjusted for field variation in citation rates, a structural understanding of citation behavior in different fields, and article recommendations which take into account article visibility and citation patterns. We develop an efficient algorithm for model fitting using variational methods. To scale up to large networks, we develop an online variant using stochastic gradient methods and case-control likelihood approximation. We apply our methods to the benchmark KDD Cup 2003 dataset with approximately 30,000 high energy physics papers.

In recent years, many methods have been developed for detecting causal relationships in observational data. Some of them have the potential to tackle large data sets. However, these methods fail to discover a combined cause, i.e. a multi-factor cause consisting of two or more component variables which individually are not causes. A straightforward approach to uncovering a combined cause is to include both individual and combined variables in the causal discovery using existing methods, but this scheme is computationally infeasible due to the huge number of combined variables. In this paper, we propose a novel approach to address this practical causal discovery problem, i.e. mining combined causes in large data sets. The experiments with both synthetic and real world data sets show that the proposed method can obtain high-quality causal discoveries with a high computational efficiency.

In a plethora of applications dealing with inverse problems, e.g. in image processing, social networks, compressive sensing, biological data processing etc., the signal of interest is known to be structured in several ways at the same time. This premise has recently guided the research to the innovative and meaningful idea of imposing multiple constraints on the parameters involved in the problem under study. For instance, when dealing with problems whose parameters form sparse and low-rank matrices, the adoption of suitably combined constraints imposing sparsity and low-rankness, is expected to yield substantially enhanced estimation results. In this paper, we address the spectral unmixing problem in hyperspectral images. Specifically, two novel unmixing algorithms are introduced, in an attempt to exploit both spatial correlation and sparse representation of pixels lying in homogeneous regions of hyperspectral images. To this end, a novel convex mixed penalty term is first defined consisting of the sum of the weighted $\ell_1$ and the weighted nuclear norm of the abundance matrix corresponding to a small area of the image determined by a sliding square window. This penalty term is then used to regularize a conventional quadratic cost function and impose simultaneously sparsity and row-rankness on the abundance matrix. The resulting regularized cost function is minimized by a) an incremental proximal sparse and low-rank unmixing algorithm and b) an algorithm based on the alternating minimization method of multipliers (ADMM). The effectiveness of the proposed algorithms is illustrated in experiments conducted both on simulated and real data.

We develop a parallel variational inference (VI) procedure for use in data-distributed settings, where each machine only has access to a subset of data and runs VI independently, without communicating with other machines. This type of "embarrassingly parallel" procedure has recently been developed for MCMC inference algorithms; however, in many cases it is not possible to directly extend this procedure to VI methods without requiring certain restrictive exponential family conditions on the form of the model. Furthermore, most existing (nonparallel) VI methods are restricted to use on conditionally conjugate models, which limits their applicability. To combat these issues, we make use of the recently proposed nonparametric VI to facilitate an embarrassingly parallel VI procedure that can be applied to a wider scope of models, including to nonconjugate models. We derive our embarrassingly parallel VI algorithm, analyze our method theoretically, and demonstrate our method empirically on a few nonconjugate models.

We study learning problems in which the conditional distribution of the output given the input varies as a function of additional task variables. In varying-coefficient models with Gaussian process priors, a Gaussian process generates the functional relationship between the task variables and the parameters of this conditional. Varying-coefficient models subsume hierarchical Bayesian multitask models, but also generalizations in which the conditional varies continuously, for instance, in time or space. However, Bayesian inference in varying-coefficient models is generally intractable. We show that inference for varying-coefficient models with isotropic Gaussian process priors resolves to standard inference for a Gaussian process that can be solved efficiently. MAP inference in this model resolves to multitask learning using task and instance kernels, and inference for hierarchical Bayesian multitask models can be carried out efficiently using graph-Laplacian kernels. We report on experiments for geospatial prediction.

This paper contains the consideration of inheritance mechanism in such knowledge representation models as object-oriented programming, frames and object-oriented dynamic networks. In addition, inheritance within representation of vague and imprecise knowledge are also discussed. New types of inheritance, general classification of all known inheritance types and approach, which allows avoiding in many cases problems with exceptions, redundancy and ambiguity within object-oriented dynamic networks and their fuzzy extension, are introduced in the paper. The proposed approach bases on conception of homogeneous and inhomogeneous or heterogeneous class of objects, which allow building of inheritance hierarchy more flexibly and efficiently.