102,836 research outputs found
Complexity of parallel matrix computations
AbstractWe estimate parallel complexity of several matrix computations under both Boolean and arithmetic machine models using deterministic and probabilistic approaches. Those computations include the evaluation of the inverse, the determinant, and the characteristic polynomial of a matrix. Recently, processor efficiency of the previous parallel algorithms for numerical matrix inversion has been substantially improved in (Pan and Reif, 1987), reaching optimum estimates up to within a logarithmic factor; that work, however, applies neither to the evaluation of the determinant and the characteristic polynomial nor to exact matrix inversion nor to the numerical inversion of ill-conditioned matrices. We present four new approaches to the solution of those latter problems (having several applications to combinatorial computations) in order to extend the suboptimum time and processor bounds of (Pan and Reif, 1987) to the case of computing the inverse, determinant, and characteristic polynomial of an arbitrary integer input matrix. In addition, processor efficient algorithms using polylogarithmic parallel time are devised for some other matrix computations, such as triangular and QR-factorizations of a matrix and its reduction to Hessenberg form
Parallelizable sparse inverse formulation Gaussian processes (SpInGP)
We propose a parallelizable sparse inverse formulation Gaussian process
(SpInGP) for temporal models. It uses a sparse precision GP formulation and
sparse matrix routines to speed up the computations. Due to the state-space
formulation used in the algorithm, the time complexity of the basic SpInGP is
linear, and because all the computations are parallelizable, the parallel form
of the algorithm is sublinear in the number of data points. We provide example
algorithms to implement the sparse matrix routines and experimentally test the
method using both simulated and real data.Comment: Presented at Machine Learning in Signal Processing (MLSP2017
Communication Lower Bounds for Distributed-Memory Computations
In this paper we propose a new approach to the study of the communication requirements of distributed computations, which advocates for the removal of the restrictive assumptions under which earlier results were derived. We illustrate our approach by giving tight lower bounds on the communication complexity required to solve several computational problems in a distributed-memory parallel machine, namely standard matrix multiplication, stencil computations, comparison sorting, and the Fast Fourier Transform. Our bounds rely only on a mild assumption on work distribution, and significantly strengthen previous results which require either the computation to be balanced among the processors, or specific initial distributions of the input data, or an upper bound on the size of processors\u27 local memories
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Parametrization of Newton's iteration for computations with structured matrices and applications
We apply a new parametrized version of Newton's iteration in order to compute (over any field F of constants) the solution, or at least-squares solution, to linear system Bx = v with an n × n Toeplitz or Toeplitz-like matrix B, as well as the determinant of B and the coefficients of its characteristic polynomial, det(λI − B), dramatically improving the processor efficiency of the known fast parallel algorithms. Our algorithms, together with some previously known and some recent results of [1““5], as well as with our new techniques for computing polynomial god's and lcm's, imply respective improvement of the known estimates for parallel arithmetic complexity of several fundamental computations with polynomials, and with both structured and general matrices
Graphs, Matrices, and the GraphBLAS: Seven Good Reasons
The analysis of graphs has become increasingly important to a wide range of
applications. Graph analysis presents a number of unique challenges in the
areas of (1) software complexity, (2) data complexity, (3) security, (4)
mathematical complexity, (5) theoretical analysis, (6) serial performance, and
(7) parallel performance. Implementing graph algorithms using matrix-based
approaches provides a number of promising solutions to these challenges. The
GraphBLAS standard (istc- bigdata.org/GraphBlas) is being developed to bring
the potential of matrix based graph algorithms to the broadest possible
audience. The GraphBLAS mathematically defines a core set of matrix-based graph
operations that can be used to implement a wide class of graph algorithms in a
wide range of programming environments. This paper provides an introduction to
the GraphBLAS and describes how the GraphBLAS can be used to address many of
the challenges associated with analysis of graphs.Comment: 10 pages; International Conference on Computational Science workshop
on the Applications of Matrix Computational Methods in the Analysis of Modern
Dat
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