The Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function (the "Bregman function"). Bregman functions and divergences have been extensively investigated during the last decades and have found applications in optimization, operations research, information theory, nonlinear analysis, machine learning and more. This paper reexamines various aspects related to the theory of Bregman functions and divergences. In particular, it presents many sufficient conditions which allow the construction of Bregman functions in a general setting and introduces new Bregman functions (such as a negative iterated log entropy). Moreover, it sheds new light on several known Bregman functions such as quadratic entropies, the negative Havrda-Charvát-Tsallis entropy, and the negative Boltzmann-Gibbs-Shannon entropy, and it shows that the negative Burg entropy, which is not a Bregman function according to the classical theory but nevertheless is known to have "Bregmanian properties", can, by our reexamination of the theory, be considered as a Bregman function. Our analysis yields several byproducts of independent interest such as the introduction of the concept of relative uniform convexity (a certain generalization of uniform convexity), new properties of uniformly and strongly convex functions, and results in Banach space theory.

Online structure learning approaches, such as those stemming from Statistical Relational Learning, enable the discovery of complex relations in noisy data streams. However, these methods assume the existence of fully-labelled training data, which is unrealistic for most real-world applications. We present a novel approach for completing the supervision of a semi-supervised structure learning task. We incorporate graph cut minimisation, a technique that derives labels for unlabelled data, based on their distance to their labelled counterparts. In order to adapt graph cut minimisation to first order logic, we employ a suitable structural distance for measuring the distance between sets of logical atoms. The labelling process is achieved online (single-pass) by means of a caching mechanism and the Hoeffding bound, a statistical tool to approximate globally-optimal decisions from locally-optimal ones. We evaluate our approach on the task of composite event recognition by using a benchmark dataset for human activity recognition, as well as a real dataset for maritime monitoring. The evaluation suggests that our approach can effectively complete the missing labels and eventually, improve the accuracy of the underlying structure learning system.

The Power Mean Laplacian for Multilayer Graph ClusteringPedro Mercado 1 Antoine Gautier 1 Francesco Tudisco 2 Matthias Hein 1 1 Department of Mathematics and Computer Science, Saarland University, Germany 2 Department of Mathematics and Statistics, University of Strathclyde, G11XH Glasgow, UK Abstract Multilayer graphs encode different kind of interactions between the same set of entities. When one wants to cluster such a multilayer graph, the natural question arises how one should merge the information from different layers. We introduce in this paper a one-parameter family of matrix power means for merging the Laplacians from different layers and analyze it in expectation in the stochastic block model. We show that this family allows to recover ground truth clusters under different settings and verify this in real world data. While computing the matrix power mean can be very expensive for large graphs, we introduce a numerical scheme to efficiently compute its eigenvectors for the ...

Embedded markup of Web pages has seen widespread adoption throughout the past years driven by standards such as RDFa and Microdata and initiatives such as schema.org, where recent studies show an adoption by 39% of all Web pages already in 2016. While this constitutes an important information source for tasks such as Web search, Web page classification or knowledge graph augmentation, individual markup nodes are usually sparsely described and often lack essential information. For instance, from 26 million nodes describing events within the Common Crawl in 2016, 59% of nodes provide less than six statements and only 257,000 nodes (0.96%) are typed with more specific event subtypes. Nevertheless, given the scale and diversity of Web markup data, nodes that provide missing information can be obtained from the Web in large quantities, in particular for categorical properties. Such data constitutes potential training data for inferring missing information to significantly augment sparsely described nodes. In this work, we introduce a supervised approach for inferring missing categorical properties in Web markup. Our experiments, conducted on properties of events and movies, show a performance of 79% and 83% F1 score correspondingly, significantly outperforming existing baselines.

Recent advances in the field of inverse reinforcement learning (IRL) have yielded sophisticated frameworks which relax the original modeling assumption that the behavior of an observed agent reflects only a single intention. Instead, the demonstration data is typically divided into parts, to account for the fact that different trajectories may correspond to different intentions, e.g., because they were generated by different domain experts. In this work, we go one step further: using the intuitive concept of subgoals, we build upon the premise that even a single trajectory can be explained more efficiently locally within a certain context than globally, enabling a more compact representation of the observed behavior. Based on this assumption, we build an implicit intentional model of the agent's goals to forecast its behavior in unobserved situations. The result is an integrated Bayesian prediction framework which provides smooth policy estimates that are consistent with the expert's plan and significantly outperform existing IRL solutions. Most notably, our framework naturally handles situations where the intentions of the agent change with time and classical IRL algorithms fail. In addition, due to its probabilistic nature, the model can be straightforwardly applied in an active learning setting to guide the demonstration process of the expert.

Data protection constraints frequently require distributed analysis of data, i.e. individual-level data remains at many different sites, but analysis nevertheless has to be performed jointly. The data exchange is often handled manually, requiring explicit permission before transfer, i.e. the number of data calls and the amount of data should be limited. Thus, only simple summary statistics are typically transferred and aggregated with just a single call, but this does not allow for complex statistical techniques, e.g., automatic variable selection for prognostic signature development. We propose a multivariable regression approach for building a prognostic signature by automatic variable selection that is based on aggregated data from different locations in iterative calls. To minimize the amount of transferred data and the number of calls, we also provide a heuristic variant of the approach. To further strengthen data protection, the approach can also be combined with a trusted third party architecture. We evaluate our proposed method in a simulation study comparing our results to the results obtained with the pooled individual data. The proposed method is seen to be able to detect covariates with true effect to a comparable extent as a method based on individual data, although the performance is moderately decreased if the number of sites is large. In a typical scenario, the heuristic decreases the number of data calls from more than 10 to 3. To make our approach widely available for application, we provide an implementation on top of the DataSHIELD framework.

We study the approximate nearest neighbour method for cost-sensitive classification on low-dimensional manifolds embedded within a high-dimensional feature space. We determine the minimax learning rates for distributions on a smooth manifold, in a cost-sensitive setting. This generalises a classic result of Audibert and Tsybakov. Building upon recent work of Chaudhuri and Dasgupta we prove that these minimax rates are attained by the approximate nearest neighbour algorithm, where neighbours are computed in a randomly projected low-dimensional space. In addition, we give a bound on the number of dimensions required for the projection which depends solely upon the reach and dimension of the manifold, combined with the regularity of the marginal.

The problem of complex data analysis is a central topic of modern statistical science and learning systems and is becoming of broader interest with the increasing prevalence of high-dimensional data. The challenge is to develop statistical models and autonomous algorithms that are able to acquire knowledge from raw data for exploratory analysis, which can be achieved through clustering techniques or to make predictions of future data via classification (i.e., discriminant analysis) techniques. Latent data models, including mixture model-based approaches are one of the most popular and successful approaches in both the unsupervised context (i.e., clustering) and the supervised one (i.e, classification or discrimination). Although traditionally tools of multivariate analysis, they are growing in popularity when considered in the framework of functional data analysis (FDA). FDA is the data analysis paradigm in which the individual data units are functions (e.g., curves, surfaces), rather than simple vectors. In many areas of application, the analyzed data are indeed often available in the form of discretized values of functions or curves (e.g., time series, waveforms) and surfaces (e.g., 2d-images, spatio-temporal data). This functional aspect of the data adds additional difficulties compared to the case of a classical multivariate (non-functional) data analysis. We review and present approaches for model-based clustering and classification of functional data. We derive well-established statistical models along with efficient algorithmic tools to address problems regarding the clustering and the classification of these high-dimensional data, including their heterogeneity, missing information, and dynamical hidden structure. The presented models and algorithms are illustrated on real-world functional data analysis problems from several application area.

We introduce a new class of graphical models that generalizes Lauritzen-Wermuth-Frydenberg chain graphs by relaxing the semi-directed acyclity constraint so that only directed cycles are forbidden. Moreover, up to two edges are allowed between any pair of nodes. Specifically, we present local, pairwise and global Markov properties for the new graphical models and prove their equivalence. We also present an equivalent factorization property. Finally, we present a causal interpretation of the new models.

A drone taxi using 5G technology is displayed at the Mobile World Congress on Feb. 27, 2018. Visitors try out Virtual 5G technology during the Mobile World Congress on Feb. 27, 2018. A 5G antenna is displayed at the Deutsche Telekom stand on the first day of the Mobile World Congress. Docomo 5G Robot remote humanoid assistant draws some'sumi-e' style drawings on the first day of the Mobile World Congress. Visitors look at a US company Qualcomm stand announcing '5G' technology at the Mobile World Congress (MWC) in Barcelona, Spain, 26 February 2018 (Photo: EPA-EFE/ALBERTO ESTEVEZ) People walk by a 5G stand at the Mobile World Congress (MWC), the world's biggest mobile fair, on February 26, 2018 in Barcelona.