We propose a novel class of statistical divergences called \textit{Relaxed Wasserstein} (RW) divergence. RW divergence generalizes Wasserstein distance and is parametrized by strictly convex, differentiable functions. We establish for RW several key probabilistic properties, which are critical for the success of Wasserstein distances. In particular, we show that RW is dominated by Total Variation (TV) and Wasserstein-$L^2$ distance, and establish continuity, differentiability, and duality representation of RW divergence. Finally, we provide a non-asymptotic moment estimate and a concentration inequality for RW divergence. Our experiments on image generation problems show that RWGANs with Kullback-Leibler (KL) divergence provide competitive performance compared with many state-of-the-art approaches. Empirically, we show that RWGANs possess better convergence properties than WGANs, with competitive inception scores. In comparison to the existing literature in GANs, which are ad-hoc in the choices of cost functions, this new conceptual framework not only provides great flexibility in designing general cost functions, e.g., for applications to GANs, but also allows different cost functions implemented and compared under a unified mathematical framework.

The time-evolving precision matrix of a piecewise-constant Gaussian graphical model encodes the dynamic conditional dependency structure of a multivariate time-series. Traditionally, graphical models are estimated under the assumption that data is drawn identically from a generating distribution. Introducing sparsity and sparse-difference inducing priors we relax these assumptions and propose a novel regularized M-estimator to jointly estimate both the graph and changepoint structure. The resulting estimator possesses the ability to therefore favor sparse dependency structures and/or smoothly evolving graph structures, as required. Moreover, our approach extends current methods to allow estimation of changepoints that are grouped across multiple dependencies in a system. An efficient algorithm for estimating structure is proposed. We study the empirical recovery properties in a synthetic setting. The qualitative effect of grouped changepoint estimation is then demonstrated by applying the method on two real-world data-sets.

Process mining is a research field focused on the analysis of event data with the aim of extracting insights related to dynamic behavior. Applying process mining techniques on data from smart home environments has the potential to provide valuable insights into (un)healthy habits and to contribute to ambient assisted living solutions. Finding the right event labels to enable the application of process mining techniques is however far from trivial, as simply using the triggering sensor as the label for sensor events results in uninformative models that allow for too much behavior (i.e., the models are overgeneralizing). Refinements of sensor level event labels suggested by domain experts have been shown to enable discovery of more precise and insightful process models. However, there exists no automated approach to generate refinements of event labels in the context of process mining. In this paper we propose a framework for the automated generation of label refinements based on the time attribute of events, allowing us to distinguish behaviourally different instances of the same event type based on their time attribute. We show on a case study with real-life smart home event data that using automatically generated refined labels in process discovery, we can find more specific, and therefore more insightful, process models. We observe that one label refinement could have an effect on the usefulness of other label refinements when used together. Therefore, we explore four strategies to generate useful combinations of multiple label refinements and evaluate those on three real-life smart home event logs.

In machine learning, there are many ways to build a product or solution and each way assumes something different. Many times, it's not obvious how to navigate and identify which assumptions are reasonable. People new to machine learning make mistakes, which in hindsight will often feel silly. I've created a list of the top mistakes that novice machine learning engineers make. Hopefully, you can learn from these common errors and create more robust solutions that bring real value.

One of the main goals in the Bitcoin analytics is price forecasting. There are many factors which influence the price dynamics. The most important factors are: the interaction between supply and demand, attractiveness for investors, financial and macroeconomics indicators, technical indicators such as difficulty, how many blocks were created recently, etc. A very important impact on the cryptocurrency price has trends in social networks and search engines. Using these factors, one can create a regression model with good fitting of bitcoin price on the historical data.

In this paper, we propose a stochastic optimization method that adaptively controls the sample size used in the computation of gradient approximations. Unlike other variance reduction techniques that either require additional storage or the regular computation of full gradients, the proposed method reduces variance by increasing the sample size as needed. The decision to increase the sample size is governed by an inner product test that ensures that search directions are descent directions with high probability. We show that the inner product test improves upon the well known norm test, and can be used as a basis for an algorithm that is globally convergent on nonconvex functions and enjoys a global linear rate of convergence on strongly convex functions. Numerical experiments on logistic regression problems illustrate the performance of the algorithm.

This paper shows that a simple baseline based on a Bag-of-Words (BoW) representation learns surprisingly good knowledge graph embeddings. By casting knowledge base completion and question answering as supervised classification problems, we observe that modeling co-occurences of entities and relations leads to state-of-the-art performance with a training time of a few minutes using the open sourced library fastText.

Nonnegative matrix factorization (NMF), a dimensionality reduction and factor analysis method, is a special case in which factor matrices have low-rank nonnegative constraints. Considering the stochastic learning in NMF, we specifically address the multiplicative update (MU) rule, which is the most popular, but which has slow convergence property. This present paper introduces on the stochastic MU rule a variance-reduced technique of stochastic gradient. Numerical comparisons suggest that our proposed algorithms robustly outperform state-of-the-art algorithms across different synthetic and real-world datasets.

Development systems for deep learning, such as Theano, Torch, TensorFlow, or MXNet, are easy-to-use tools for creating complex neural network models. Since gradient computations are automatically baked in, and execution is mapped to high performance hardware, these models can be trained end-to-end on large amounts of data. However, it is currently not easy to implement many basic machine learning primitives in these systems (such as Gaussian processes, least squares estimation, principal components analysis, Kalman smoothing), mainly because they lack efficient support of linear algebra primitives as differentiable operators. We detail how a number of matrix decompositions (Cholesky, LQ, symmetric eigen) can be implemented as differentiable operators. We have implemented these primitives in MXNet, running on CPU and GPU in single and double precision. We sketch use cases of these new operators, learning Gaussian process and Bayesian linear regression models. Our implementation is based on BLAS/LAPACK APIs, for which highly tuned implementations are available on all major CPUs and GPUs.

Datasets with a mixture of categorical and numerical attributes are pervasive in applications from business and socioeconomic settings. Clustering these datasets is an important activity in their analysis. Techniques to cluster these datasets have been developed by researchers, see for example [1], [2] and [3]. Techniques to cluster mixed datasets either prescribe a probabilistic generative model [4] or use a dissimilarity measure [5] to compute a dissimilarity matrix that is then clustered. Each of these approaches have issues that need to be addressed when they are applied to big datasets - datasets with a large number of instances compared to attributes.