Publications

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Network science is a powerful tool for analyzing complex systems in fields ranging from sociology to engineering to biology. This paper is focused on generative models of bipartite graphs, also known as two-way graphs. We propose two generative models that can be easily tuned to reproduce the characteristics of real-world networks, not just qualitatively, but quantitatively. The measurements we consider are the degree distributions and the bipartite clustering coefficient, which we refer to as the metamorphosis coefficient. We define edge, node, and degreewise metamorphosis coefficients, enabling a more detailed understand of the bipartite community structure. Our proposed bipartite Chung-Lu model is able to reproduce real-world degree distributions, and our proposed bipartite “BTER” model reproduces both the degree distributions as well as the degreewise metamorphosis coefficients. We demonstrate the effectiveness of these models on several real-world data sets.

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As parallel computing trends towards the exascale, scientific data produced by high-fidelity simulations are growing increasingly massive. For instance, a simulation on a three-dimensional spatial grid with 512 points per dimension that tracks 64 variables per grid point for 128 time steps yields 8 TB of data. By viewing the data as a dense five way tensor, we can compute a Tucker decomposition to find inherent low-dimensional multilinear structure, achieving compression ratios of up to 10000 on real-world data sets with negligible loss in accuracy. So that we can operate on such massive data, we present the first-ever distributed memory parallel implementation for the Tucker decomposition, whose key computations correspond to parallel linear algebra operations, albeit with nonstandard data layouts. Our approach specifies a data distribution for tensors that avoids any tensor data redistribution, either locally or in parallel. We provide accompanying analysis of the computation and communication costs of the algorithms. To demonstrate the compression and accuracy of the method, we apply our approach to real-world data sets from combustion science simulations. We also provide detailed performance results, including parallel performance in both weak and strong scaling experiments.

Large-scale datasets in computational chemistry typically require distributed-memory parallel methods to perform a special operation known as tensor contraction. Tensors are multidimensional arrays, and a tensor contraction is akin to matrix multiplication with special types of permutations. Creating an efficient algorithm and optimized im- plementation in this domain is complex, tedious, and error-prone. To address this, we develop a notation to express data distributions so that we can apply use automated methods to find optimized implementations for tensor contractions. We consider the spin-adapted coupled cluster singles and doubles method from computational chemistry and use our methodology to produce an efficient implementation. Experiments per- formed on the IBM Blue Gene/Q and Cray XC30 demonstrate impact both improved performance and reduced memory consumption.

We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued vectors. Algebraic methods exist for computing complex-valued decompositions of symmetric tensors, but here we focus on real-valued decompositions, both unconstrained and nonnegative, for problems with low-rank structure. We discuss when solutions exist and how to formulate the mathematical program. Numerical results show the properties of the proposed formulations (including one that ignores symmetry) on a set of test problems and illustrate that these straightforward formulations can be effective even though the problem is nonconvex.

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Tensor factorizations with nonnegativity constraints have found application in analysing data from cyber traffic, social networks, and other areas. We consider application data best described as being generated by a Poisson process (e.g. count data), which leads to sparse tensors that can be modelled by sparse factor matrices. In this paper, we investigate efficient techniques for computing an appropriate canonical polyadic tensor factorization based on the Kullback–Leibler divergence function. We propose novel subproblem solvers within the standard alternating block variable approach. Our new methods exploit structure and reformulate the optimization problem as small independent subproblems. We employ bound-constrained Newton and quasi-Newton methods. Finally, we compare our algorithms against other codes, demonstrating superior speed for high accuracy results and the ability to quickly find sparse solutions.

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