Abstract-In a variety of research areas, the weighted bag of vectors and the histogram are widely used descriptors for complex objects. Both can be expressed as discrete distributions. D2-clustering pursues the minimum total within-cluster variation for a set of discrete distributions subject to the KantorovichWasserstein metric. D2-clustering has a severe scalability issue, the bottleneck being the computation of a centroid distribution, called Wasserstein barycenter, that minimizes its sum of squared distances to the cluster members. In this paper, we develop a modified Bregman ADMM approach for computing the approximate discrete Wasserstein barycenter of large clusters. In the case when the support points of the barycenters are unknown and have low cardinality, our method achieves high accuracy empirically at a much reduced computational cost. The strengths and weaknesses of our method and its alternatives are examined through experiments, and we recommend scenarios for their respective usage. Moreover, we develop both serial and parallelized versions of the algorithm. By experimenting with large-scale data, we demonstrate the computational efficiency of the new methods and investigate their convergence properties and numerical stability. The clustering results obtained on several datasets in different domains are highly competitive in comparison with some widely used methods in the corresponding areas.

Although convolutional neural networks (CNNs) can be used to classify electrocardiogram (ECG) beats in the diagnosis of cardiovascular disease, ECG signals are typically processed as one-dimensional signals while CNNs are better suited to multidimensional pattern or image recognition applications. In this study, the morphology and rhythm of heartbeats are fused into a two-dimensional information vector for subsequent processing by CNNs that include adaptive learning rate and biased dropout methods. The results demonstrate that the proposed CNN model is effective for detecting irregular heartbeats or arrhythmias via automatic feature extraction. When the proposed model was tested on the MIT-BIH arrhythmia database, the model achieved higher performance than other state-of-the-art methods for five and eight heartbeat categories (the average accuracy was 99.1% and 97%). In particular, the proposed system had better performance in terms of the sensitivity and positive predictive rate for V beats by more than 4.3% and 5.4%, respectively, and also for S beats by more than 22.6% and 25.9%, respectively, when compared to existing algorithms. It is anticipated that the proposed method will be suitable for implementation on portable devices for the e-home health monitoring of cardiovascular disease.

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