On Optimizing Distributed Tucker Decomposition for Dense Tensors

The Tucker decomposition expresses a given tensor as the product of a small core tensor and a set of factor matrices. Apart from providing data compression, the construction is useful in performing analysis such as principal component analysis (PCA)and finds applications in diverse domains such as signal processing, computer vision and text analytics. Our objective is to develop an efficient distributed implementation for the case of dense tensors. The implementation is based on the HOOI (Higher Order Orthogonal Iterator) procedure, wherein the tensor-times-matrix product forms the core routine. Prior work have proposed heuristics for reducing the computational load and communication volume incurred by the routine. We study the two metrics in a formal and systematic manner, and design strategies that are optimal under the two fundamental metrics. Our experimental evaluation on a large benchmark of tensors shows that the optimal strategies provide significant reduction in load and volume compared to prior heuristics, and provide up to 7x speed-up in the overall running time.Comment: Preliminary version of the paper appears in the proceedings of IPDPS'1

GraphBLAST: A High-Performance Linear Algebra-based Graph Framework on the GPU

High-performance implementations of graph algorithms are challenging to implement on new parallel hardware such as GPUs because of three challenges: (1) the difficulty of coming up with graph building blocks, (2) load imbalance on parallel hardware, and (3) graph problems having low arithmetic intensity. To address some of these challenges, GraphBLAS is an innovative, on-going effort by the graph analytics community to propose building blocks based on sparse linear algebra, which will allow graph algorithms to be expressed in a performant, succinct, composable and portable manner. In this paper, we examine the performance challenges of a linear-algebra-based approach to building graph frameworks and describe new design principles for overcoming these bottlenecks. Among the new design principles is exploiting input sparsity, which allows users to write graph algorithms without specifying push and pull direction. Exploiting output sparsity allows users to tell the backend which values of the output in a single vectorized computation they do not want computed. Load-balancing is an important feature for balancing work amongst parallel workers. We describe the important load-balancing features for handling graphs with different characteristics. The design principles described in this paper have been implemented in "GraphBLAST", the first high-performance linear algebra-based graph framework on NVIDIA GPUs that is open-source. The results show that on a single GPU, GraphBLAST has on average at least an order of magnitude speedup over previous GraphBLAS implementations SuiteSparse and GBTL, comparable performance to the fastest GPU hardwired primitives and shared-memory graph frameworks Ligra and Gunrock, and better performance than any other GPU graph framework, while offering a simpler and more concise programming model.Comment: 50 pages, 14 figures, 14 table

Polynomial Chaos Expansion of random coefficients and the solution of stochastic partial differential equations in the Tensor Train format

We apply the Tensor Train (TT) decomposition to construct the tensor product Polynomial Chaos Expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format, but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. Besides, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its post-processing, staying in the TT format. We compare our technique with the traditional sparse polynomial chaos and the Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial degree is bounded for each random variable independently. This provides higher accuracy than the sparse polynomial set or the Monte Carlo method, but the cardinality of the tensor product set grows exponentially with the number of random variables. However, when the PCE coefficients are implicitly approximated in the TT format, the computations with the full tensor product polynomial set become possible. In the numerical experiments, we confirm that the new methodology is competitive in a wide range of parameters, especially where high accuracy and high polynomial degrees are required.Comment: This is a major revision of the manuscript arXiv:1406.2816 with significantly extended numerical experiments. Some unused material is remove