randUTV: A blocked randomized algorithm for computing a rank-revealing UTV factorization

This manuscript describes the randomized algorithm randUTV for computing a so called UTV factorization efficiently. Given a matrix $A$, the algorithm computes a factorization $A = UTV^{*}$, where $U$ and $V$ have orthonormal columns, and $T$ is triangular (either upper or lower, whichever is preferred). The algorithm randUTV is developed primarily to be a fast and easily parallelized alternative to algorithms for computing the Singular Value Decomposition (SVD). randUTV provides accuracy very close to that of the SVD for problems such as low-rank approximation, solving ill-conditioned linear systems, determining bases for various subspaces associated with the matrix, etc. Moreover, randUTV produces highly accurate approximations to the singular values of $A$. Unlike the SVD, the randomized algorithm proposed builds a UTV factorization in an incremental, single-stage, and non-iterative way, making it possible to halt the factorization process once a specified tolerance has been met. Numerical experiments comparing the accuracy and speed of randUTV to the SVD are presented. These experiments demonstrate that in comparison to column pivoted QR, which is another factorization that is often used as a relatively economic alternative to the SVD, randUTV compares favorably in terms of speed while providing far higher accuracy

Online Tensor Methods for Learning Latent Variable Models

We introduce an online tensor decomposition based approach for two latent variable modeling problems namely, (1) community detection, in which we learn the latent communities that the social actors in social networks belong to, and (2) topic modeling, in which we infer hidden topics of text articles. We consider decomposition of moment tensors using stochastic gradient descent. We conduct optimization of multilinear operations in SGD and avoid directly forming the tensors, to save computational and storage costs. We present optimized algorithm in two platforms. Our GPU-based implementation exploits the parallelism of SIMD architectures to allow for maximum speed-up by a careful optimization of storage and data transfer, whereas our CPU-based implementation uses efficient sparse matrix computations and is suitable for large sparse datasets. For the community detection problem, we demonstrate accuracy and computational efficiency on Facebook, Yelp and DBLP datasets, and for the topic modeling problem, we also demonstrate good performance on the New York Times dataset. We compare our results to the state-of-the-art algorithms such as the variational method, and report a gain of accuracy and a gain of several orders of magnitude in the execution time.Comment: JMLR 201

Using reconfigurable computing technology to accelerate matrix decomposition and applications

Matrix decomposition plays an increasingly significant role in many scientific and engineering applications. Among numerous techniques, Singular Value Decomposition (SVD) and Eigenvalue Decomposition (EVD) are widely used as factorization tools to perform Principal Component Analysis for dimensionality reduction and pattern recognition in image processing, text mining and wireless communications, while QR Decomposition (QRD) and sparse LU Decomposition (LUD) are employed to solve the dense or sparse linear system of equations in bioinformatics, power system and computer vision. Matrix decompositions are computationally expensive and their sequential implementations often fail to meet the requirements of many time-sensitive applications. The emergence of reconfigurable computing has provided a flexible and low-cost opportunity to pursue high-performance parallel designs, and the use of FPGAs has shown promise in accelerating this class of computation. In this research, we have proposed and implemented several highly parallel FPGA-based architectures to accelerate matrix decompositions and their applications in data mining and signal processing. Specifically, in this dissertation we describe the following contributions: • We propose an efficient FPGA-based double-precision floating-point architecture for EVD, which can efficiently analyze large-scale matrices. • We implement a floating-point Hestenes-Jacobi architecture for SVD, which is capable of analyzing arbitrary sized matrices. • We introduce a novel deeply pipelined reconfigurable architecture for QRD, which can be dynamically configured to perform either Householder transformation or Givens rotation in a manner that takes advantage of the strengths of each. • We design a configurable architecture for sparse LUD that supports both symmetric and asymmetric sparse matrices with arbitrary sparsity patterns. • By further extending the proposed hardware solution for SVD, we parallelize a popular text mining tool-Latent Semantic Indexing with an FPGA-based architecture. • We present a configurable architecture to accelerate Homotopy l1-minimization, in which the modification of the proposed FPGA architecture for sparse LUD is used at its core to parallelize both Cholesky decomposition and rank-1 update. Our experimental results using an FPGA-based acceleration system indicate the efficiency of our proposed novel architectures, with application and dimension-dependent speedups over an optimized software implementation that range from 1.5ÃÂ to 43.6ÃÂ in terms of computation time

Executing linear algebra kernels in heterogeneous distributed infrastructures with PyCOMPSs

Python is a popular programming language due to the simplicity of its syntax, while still achieving a good performance even being an interpreted language. The adoption from multiple scientific communities has evolved in the emergence of a large number of libraries and modules, which has helped to put Python on the top of the list of the programming languages [1]. Task-based programming has been proposed in the recent years as an alternative parallel programming model. PyCOMPSs follows such approach for Python, and this paper presents its extensions to combine task-based parallelism and thread-level parallelism. Also, we present how PyCOMPSs has been adapted to support heterogeneous architectures, including Xeon Phi and GPUs. Results obtained with linear algebra benchmarks demonstrate that significant performance can be obtained with a few lines of Python.This work has been supported by the Spanish Government (SEV2015-0493), by the Spanish Ministry of Science and Innovation (contract TIN2015-65316-P), by Generalitat de Catalunya (contracts 2014-SGR-1051 and 2014-SGR-1272). Javier Conejero postdoctoral contract is co-financed by the Ministry of Economy and Competitiveness under Juan de la Cierva Formación postdoctoral fellowship number FJCI-2015-24651. Cristian Ramon-Cortes predoctoral contract is financed by the Ministry of Economy and Competitiveness under the contract BES-2016-076791. This work is supported by the Intel-BSC Exascale Lab. This work has been supported by the European Commission through the Horizon 2020 Research and Innovation program under contract 687584 (TANGO project).Peer ReviewedPostprint (published version

Accelerating Industrial Applications: The Development of Basic GPU Kernels for the New Block AMG Algorithms for Solving SLE with Explicitly Calculated Sparse Basis

AbstractNowadays, GPU computations are playing significant role in supercomputing technologies. This work is a part of a project dealing with solving problems of modeling hydro- and aerodynamics where linear algebra operations are frequently used and occupy most of execution time. In despite of the fact that GPUs are traditionally used for solving high sized problems, in our project we need to solve many tasks of low sizes. Because of this, modern library's solutions such as cuBLAS (1) and cuSPARSE (2) are not suitable enough for that, so we have a task of implementation more efficient functions for concrete linear algebra operations taking into account its specialties