Algorithmic patterns for H\mathcal{H}-matrices on many-core processors

In this work, we consider the reformulation of hierarchical ($\mathcal{H}$) matrix algorithms for many-core processors with a model implementation on graphics processing units (GPUs). $\mathcal{H}$ matrices approximate specific dense matrices, e.g., from discretized integral equations or kernel ridge regression, leading to log-linear time complexity in dense matrix-vector products. The parallelization of $\mathcal{H}$ matrix operations on many-core processors is difficult due to the complex nature of the underlying algorithms. While previous algorithmic advances for many-core hardware focused on accelerating existing $\mathcal{H}$ matrix CPU implementations by many-core processors, we here aim at totally relying on that processor type. As main contribution, we introduce the necessary parallel algorithmic patterns allowing to map the full $\mathcal{H}$ matrix construction and the fast matrix-vector product to many-core hardware. Here, crucial ingredients are space filling curves, parallel tree traversal and batching of linear algebra operations. The resulting model GPU implementation hmglib is the, to the best of the authors knowledge, first entirely GPU-based Open Source $\mathcal{H}$ matrix library of this kind. We conclude this work by an in-depth performance analysis and a comparative performance study against a standard $\mathcal{H}$ matrix library, highlighting profound speedups of our many-core parallel approach

Sequential Changepoint Approach for Online Community Detection

We present new algorithms for detecting the emergence of a community in large networks from sequential observations. The networks are modeled using Erdos-Renyi random graphs with edges forming between nodes in the community with higher probability. Based on statistical changepoint detection methodology, we develop three algorithms: the Exhaustive Search (ES), the mixture, and the Hierarchical Mixture (H-Mix) methods. Performance of these methods is evaluated by the average run length (ARL), which captures the frequency of false alarms, and the detection delay. Numerical comparisons show that the ES method performs the best; however, it is exponentially complex. The mixture method is polynomially complex by exploiting the fact that the size of the community is typically small in a large network. However, it may react to a group of active edges that do not form a community. This issue is resolved by the H-Mix method, which is based on a dendrogram decomposition of the network. We present an asymptotic analytical expression for ARL of the mixture method when the threshold is large. Numerical simulation verifies that our approximation is accurate even in the non-asymptotic regime. Hence, it can be used to determine a desired threshold efficiently. Finally, numerical examples show that the mixture and the H-Mix methods can both detect a community quickly with a lower complexity than the ES method.Comment: Submitted to 2014 INFORMS Workshop on Data Mining and Analytics and an IEEE journa

A DC Programming Approach for Solving Multicast Network Design Problems via the Nesterov Smoothing Technique

This paper continues our effort initiated in [9] to study Multicast Communication Networks, modeled as bilevel hierarchical clustering problems, by using mathematical optimization techniques. Given a finite number of nodes, we consider two different models of multicast networks by identifying a certain number of nodes as cluster centers, and at the same time, locating a particular node that serves as a total center so as to minimize the total transportation cost through the network. The fact that the cluster centers and the total center have to be among the given nodes makes this problem a discrete optimization problem. Our approach is to reformulate the discrete problem as a continuous one and to apply Nesterov smoothing approximation technique on the Minkowski gauges that are used as distance measures. This approach enables us to propose two implementable DCA-based algorithms for solving the problems. Numerical results and practical applications are provided to illustrate our approach