In this paper we introduce a novel online time series forecasting model we refer to as the pM-GP filter. We show that our model is equivalent to Gaussian process regression, with the advantage that both online forecasting and online learning of the hyper-parameters have a constant (rather than cubic) time complexity and a constant (rather than squared) memory requirement in the number of observations, without resorting to approximations. Moreover, the proposed model is expressive in that the family of covariance functions of the implied latent process, namely the spectral Matern kernels, have recently been proven to be capable of approximating arbitrarily well any translation-invariant covariance function. The benefit of our approach compared to competing models is demonstrated using experiments on several real-life datasets.

In this work, we consider the class of multi-state autoregressive processes that can be used to model non-stationary time-series of interest. In order to capture different autoregressive (AR) states underlying an observed time series, it is crucial to select the appropriate number of states. We propose a new model selection technique based on the Gap statistics, which uses a null reference distribution on the stable AR filters to check whether adding a new AR state significantly improves the performance of the model. To that end, we define a new distance measure between AR filters based on mean squared prediction error (MSPE), and propose an efficient method to generate random stable filters that are uniformly distributed in the coefficient space. Numerical results are provided to evaluate the performance of the proposed approach.

Multilayer networks are a useful data structure for simultaneously capturing multiple types of relationships between a set of nodes. In such networks, each relational definition gives rise to a layer. While each layer provides its own set of information, community structure across layers can be collectively utilized to discover and quantify underlying relational patterns between nodes. To concisely extract information from a multilayer network, we propose to identify and combine sets of layers with meaningful similarities in community structure. In this paper, we describe the "strata multilayer stochastic block model'' (sMLSBM), a probabilistic model for multilayer community structure. The central extension of the model is that there exist groups of layers, called "strata'', which are defined such that all layers in a given stratum have community structure described by a common stochastic block model (SBM). That is, layers in a stratum exhibit similar node-to-community assignments and SBM probability parameters. Fitting the sMLSBM to a multilayer network provides a joint clustering that yields node-to-community and layer-to-stratum assignments, which cooperatively aid one another during inference. We describe an algorithm for separating layers into their appropriate strata and an inference technique for estimating the SBM parameters for each stratum. We demonstrate our method using synthetic networks and a multilayer network inferred from data collected in the Human Microbiome Project.

Currently, Markov-Gibbs random field (MGRF) image models which include high-order interactions are almost always built by modelling responses of a stack of local linear filters. Actual interaction structure is specified implicitly by the filter coefficients. In contrast, we learn an explicit high-order MGRF structure by considering the learning process in terms of general exponential family distributions nested over base models, so that potentials added later can build on previous ones. We relatively rapidly add new features by skipping over the costly optimisation of parameters. We introduce the use of local binary patterns as features in MGRF texture models, and generalise them by learning offsets to the surrounding pixels. These prove effective as high-order features, and are fast to compute. Several schemes for selecting high-order features by composition or search of a small subclass are compared. Additionally we present a simple modification of the maximum likelihood as a texture modelling-specific objective function which aims to improve generalisation by local windowing of statistics. The proposed method was experimentally evaluated by learning high-order MGRF models for a broad selection of complex textures and then performing texture synthesis, and succeeded on much of the continuum from stochastic through irregularly structured to near-regular textures. Learning interaction structure is very beneficial for textures with large-scale structure, although those with complex irregular structure still provide difficulties. The texture models were also quantitatively evaluated on two tasks and found to be competitive with other works: grading of synthesised textures by a panel of observers; and comparison against several recent MGRF models by evaluation on a constrained inpainting task.

This paper considers a Bayesian view for estimating a sub-network in a Markov random field. The sub-network corresponds to the Markov blanket of a set of query variables, where the set of potential neighbours here is big. We factorize the posterior such that the Markov blanket is conditionally independent of the network of the potential neighbours. By exploiting this blockwise decoupling, we derive analytic expressions for posterior conditionals. Subsequently, we develop an inference scheme which makes use of the factorization. As a result, estimation of a sub-network is possible without inferring an entire network. Since the resulting Gibbs sampler scales linearly with the number of variables, it can handle relatively large neighbourhoods. The proposed scheme results in faster convergence and superior mixing of the Markov chain than existing Bayesian network estimation techniques.

Critically, such dimensions are uncovered based on user feedback, often in implicit form (such as purchase histories, browsing logs, etc.); in addition, some recommender systems make use of side information, such as product attributes, temporal information, or review text. However one important feature that is typically ignored by existing personalized recommendation and ranking methods is the visual appearance of the items being considered. In this paper we propose a scalable factorization model to incorporate visual signals into predictors of people's opinions, which we apply to a selection of large, real-world datasets. We make use of visual features extracted from product images using (pre-trained) deep networks, on top of which we learn an additional layer that uncovers the visual dimensions that best explain the variation in people's feedback. This not only leads to significantly more accurate personalized ranking methods, but also helps to alleviate cold start issues, and qualitatively to analyze the visual dimensions that influence people's opinions.

The task of determining labels of all network nodes based on the knowledge about network structure and labels of some training subset of nodes is called the within-network classification. It may happen that none of the labels of the nodes is known and additionally there is no information about number of classes to which nodes can be assigned. In such a case a subset of nodes has to be selected for initial label acquisition. The question that arises is: "labels of which nodes should be collected and used for learning in order to provide the best classification accuracy for the whole network?". Active learning and inference is a practical framework to study this problem. A set of methods for active learning and inference for within network classification is proposed and validated. The utility score calculation for each node based on network structure is the first step in the process. The scores enable to rank the nodes. Based on the ranking, a set of nodes, for which the labels are acquired, is selected (e.g. by taking top or bottom N from the ranking). The new measure-neighbour methods proposed in the paper suggest not obtaining labels of nodes from the ranking but rather acquiring labels of their neighbours. The paper examines 29 distinct formulations of utility score and selection methods reporting their impact on the results of two collective classification algorithms: Iterative Classification Algorithm and Loopy Belief Propagation. We advocate that the accuracy of presented methods depends on the structural properties of the examined network. We claim that measure-neighbour methods will work better than the regular methods for networks with higher clustering coefficient and worse than regular methods for networks with low clustering coefficient. According to our hypothesis, based on clustering coefficient we are able to recommend appropriate active learning and inference method.

This paper examines the role and efficiency of the non-convex loss functions for binary classification problems. In particular, we investigate how to design a simple and effective boosting algorithm that is robust to the outliers in the data. The analysis of the role of a particular non-convex loss for prediction accuracy varies depending on the diminishing tail properties of the gradient of the loss -- the ability of the loss to efficiently adapt to the outlying data, the local convex properties of the loss and the proportion of the contaminated data. In order to use these properties efficiently, we propose a new family of non-convex losses named $\gamma$-robust losses. Moreover, we present a new boosting framework, {\it Arch Boost}, designed for augmenting the existing work such that its corresponding classification algorithm is significantly more adaptable to the unknown data contamination. Along with the Arch Boosting framework, the non-convex losses lead to the new class of boosting algorithms, named adaptive, robust, boosting (ARB). Furthermore, we present theoretical examples that demonstrate the robustness properties of the proposed algorithms. In particular, we develop a new breakdown point analysis and a new influence function analysis that demonstrate gains in robustness. Moreover, we present new theoretical results, based only on local curvatures, which may be used to establish statistical and optimization properties of the proposed Arch boosting algorithms with highly non-convex loss functions. Extensive numerical calculations are used to illustrate these theoretical properties and reveal advantages over the existing boosting methods when data exhibits a number of outliers.

We consider the task of fitting a regression model involving interactions among a potentially large set of covariates, in which we wish to enforce strong heredity. We propose FAMILY, a very general framework for this task. Our proposal is a generalization of several existing methods, such as VANISH [Radchenko and James, 2010], hierNet [Bien et al., 2013], the all-pairs lasso, and the lasso using only main effects. It can be formulated as the solution to a convex optimization problem, which we solve using an efficient alternating directions method of multipliers (ADMM) algorithm. This algorithm has guaranteed convergence to the global optimum, can be easily specialized to any convex penalty function of interest, and allows for a straightforward extension to the setting of generalized linear models. We derive an unbiased estimator of the degrees of freedom of FAMILY, and explore its performance in a simulation study and on an HIV sequence data set.

We introduce an online tensor decomposition based approach for two latent variable modeling problems namely, (1) community detection, in which we learn the latent communities that the social actors in social networks belong to, and (2) topic modeling, in which we infer hidden topics of text articles. We consider decomposition of moment tensors using stochastic gradient descent. We conduct optimization of multilinear operations in SGD and avoid directly forming the tensors, to save computational and storage costs. We present optimized algorithm in two platforms. Our GPU-based implementation exploits the parallelism of SIMD architectures to allow for maximum speed-up by a careful optimization of storage and data transfer, whereas our CPU-based implementation uses efficient sparse matrix computations and is suitable for large sparse datasets. For the community detection problem, we demonstrate accuracy and computational efficiency on Facebook, Yelp and DBLP datasets, and for the topic modeling problem, we also demonstrate good performance on the New York Times dataset. We compare our results to the state-of-the-art algorithms such as the variational method, and report a gain of accuracy and a gain of several orders of magnitude in the execution time.