Pipelined ring algorithm for matrix multiplication on transputer networks: performance analysis and estimation

A parallel matrix multiplication algorithm is presented, and studies of its performance and estimation are discussed. The algorithm is implemented on a network of transputers connected in a ring topology. An efficient scheme for partitioning the input matrices is introduced which enables overlapping computation with communication. This makes the algorithm achieve near-ideal speed-up for reasonably large matrices. Analytical expressions for the execution time of the algorithm have been derived by analysing its computation and communication characteristics. These expressions are validated by comparing the theoretical results of the performance with the experimental values obtained on a four-transputer network for both square and irregular matrices. The analytical model is also used to estimate the performance of the algorithm for a varying number of transputers and varying problem sizes. Although the algorithm is implemented on transputers, the methodology and the partitioning scheme presented in this paper are quite general and can be implemented on other processors which have the capability of overlapping computation with communication. The equations for performance prediction can also be extended to other multiprocessor systems.

Sparse matrix multiplication on a reconfigurable many-core architecture

Sparse matrix-vector multiplication (SMVM) is a fundamental operation in many scientific and engineering applications. In many cases sparse matrices have thousands of rows and columns where most of the entries are zero, while non-zero data is spread over the matrix. This sparsity of data locality reduces the effectiveness of data cache in general-purpose processors quite reducing their performance efficiency when compared to what is achieved with dense matrix multiplication. In this paper, we propose a parallel processing solution for SMVM in a many-core architecture. The architecture is tested with known benchmarks using a ZYNQ-7020 FPGA. The architecture is scalable in the number of core elements and limited only by the available memory bandwidth. It achieves performance efficiencies up to almost 70% and better performances than previous FPGA designs.

Performance of an embedded optical vector matrix multiplication processor architecture

An embedded architecture of optical vector matrix multiplier (OVMM) is presented. The embedded architecture is aimed at optimising the data flow of vector matrix multiplier (VMM) to promote its performance. Data dependence is discussed when the OVMM is connected to a cluster system. A simulator is built to analyse the performance according to the architecture. According to the simulation, Amdahl's law is used to analyse the hybrid opto-electronic system. It is found that the electronic part and its interaction with optical part form the bottleneck of system.

Algorithm-oriented design of efficient many-core architectures applied to dense matrix multiplication

Recent integrated circuit technologies have opened the possibility to design parallel architectures with hundreds of cores on a single chip. The design space of these parallel architectures is huge with many architectural options. Exploring the design space gets even more difficult if, beyond performance and area, we also consider extra metrics like performance and area efficiency, where the designer tries to design the architecture with the best performance per chip area and the best sustainable performance. In this paper we present an algorithm-oriented approach to design a many-core architecture. Instead of doing the design space exploration of the many core architecture based on the experimental execution results of a particular benchmark of algorithms, our approach is to make a formal analysis of the algorithms considering the main architectural aspects and to determine how each particular architectural aspect is related to the performance of the architecture when running an algorithm or set of algorithms. The architectural aspects considered include the number of cores, the local memory available in each core, the communication bandwidth between the many-core architecture and the external memory and the memory hierarchy. To exemplify the approach we did a theoretical analysis of a dense matrix multiplication algorithm and determined an equation that relates the number of execution cycles with the architectural parameters. Based on this equation a many-core architecture has been designed. The results obtained indicate that a 100 mm(2) integrated circuit design of the proposed architecture, using a 65 nm technology, is able to achieve 464 GFLOPs (double precision floating-point) for a memory bandwidth of 16 GB/s. This corresponds to a performance efficiency of 71 %. Considering a 45 nm technology, a 100 mm(2) chip attains 833 GFLOPs which corresponds to 84 % of peak performance These figures are better than those obtained by previous many-core architectures, except for the area efficiency which is limited by the lower memory bandwidth considered. The results achieved are also better than those of previous state-of-the-art many-cores architectures designed specifically to achieve high performance for matrix multiplication.

Genetic algorithms for discovery of matrix multiplication methods

We present a parallel genetic algorithm for nding matrix multiplication algo-rithms. For 3 x 3 matrices our genetic algorithm successfully discovered algo-rithms requiring 23 multiplications, which are equivalent to the currently best known human-developed algorithms. We also studied the cases with less mul-tiplications and evaluated the suitability of the methods discovered. Although our evolutionary method did not reach the theoretical lower bound it led to an approximate solution for 22 multiplications.

Matrix-vector multiplication and triangular linear solver using GPGPU for symmetric positive definite matrices derived from elliptic equations

The modern GPUs are well suited for intensive computational tasks and massive parallel computation. Sparse matrix multiplication and linear triangular solver are the most important and heavily used kernels in scientific computation, and several challenges in developing a high performance kernel with the two modules is investigated. The main interest it to solve linear systems derived from the elliptic equations with triangular elements. The resulting linear system has a symmetric positive definite matrix. The sparse matrix is stored in the compressed sparse row (CSR) format. It is proposed a CUDA algorithm to execute the matrix vector multiplication using directly the CSR format. A dependence tree algorithm is used to determine which variables the linear triangular solver can determine in parallel. To increase the number of the parallel threads, a coloring graph algorithm is implemented to reorder the mesh numbering in a pre-processing phase. The proposed method is compared with parallel and serial available libraries. The results show that the proposed method improves the computation cost of the matrix vector multiplication. The pre-processing associated with the triangular solver needs to be executed just once in the proposed method. The conjugate gradient method was implemented and showed similar convergence rate for all the compared methods. The proposed method showed significant smaller execution time.

Matrix Decomposition Methods for Data Mining : Computational Complexity and Algorithms

Matrix decompositions, where a given matrix is represented as a product of two other matrices, are regularly used in data mining. Most matrix decompositions have their roots in linear algebra, but the needs of data mining are not always those of linear algebra. In data mining one needs to have results that are interpretable -- and what is considered interpretable in data mining can be very different to what is considered interpretable in linear algebra. --- The purpose of this thesis is to study matrix decompositions that directly address the issue of interpretability. An example is a decomposition of binary matrices where the factor matrices are assumed to be binary and the matrix multiplication is Boolean. The restriction to binary factor matrices increases interpretability -- factor matrices are of the same type as the original matrix -- and allows the use of Boolean matrix multiplication, which is often more intuitive than normal matrix multiplication with binary matrices. Also several other decomposition methods are described, and the computational complexity of computing them is studied together with the hardness of approximating the related optimization problems. Based on these studies, algorithms for constructing the decompositions are proposed. Constructing the decompositions turns out to be computationally hard, and the proposed algorithms are mostly based on various heuristics. Nevertheless, the algorithms are shown to be capable of finding good results in empirical experiments conducted with both synthetic and real-world data.

Dynamic matrix rank with partial lookahead

We consider the problem of maintaining information about the rank of a matrix $M$ under changes to its entries. For an $n \times n$ matrix $M$, we show an amortized upper bound of $O(n^{\omega-1})$ arithmetic operations per change for this problem, where $\omega < 2.376$ is the exponent for matrix multiplication, under the assumption that there is a {\em lookahead} of up to $\Theta(n)$ locations. That is, we know up to the next $\Theta(n)$ locations $(i_1,j_1),(i_2,j_2),\ldots,$ whose entries are going to change, in advance; however we do not know the new entries in these locations in advance. We get the new entries in these locations in a dynamic manner.

Fast acoustic likelihood computation using low-rank matrix approximation

Acoustic modeling using mixtures of multivariate Gaussians is the prevalent approach for many speech processing problems. Computing likelihoods against a large set of Gaussians is required as a part of many speech processing systems and it is the computationally dominant phase for LVCSR systems. We express the likelihood computation as a multiplication of matrices representing augmented feature vectors and Gaussian parameters. The computational gain of this approach over traditional methods is by exploiting the structure of these matrices and efficient implementation of their multiplication.In particular, we explore direct low-rank approximation of the Gaussian parameter matrix and indirect derivation of low-rank factors of the Gaussian parameter matrix by optimum approximation of the likelihood matrix. We show that both the methods lead to similar speedups but the latter leads to far lesser impact on the recognition accuracy. Experiments on a 1138 word vocabulary RM1 task using Sphinx 3.7 system show that, for a typical case the matrix multiplication approach leads to overall speedup of 46%. Both the low-rank approximation methods increase the speedup to around 60%, with the former method increasing the word error rate (WER) from 3.2% to 6.6%, while the latter increases the WER from 3.2% to 3.5%.

Custom coprocessor based matrix algorithms for image and signal processing

Matrix algorithms are important in many types of applications including image and signal processing. A close examination of the algorithms used in these, and related, applications reveals that many of the fundamental actions involve matrix algorithms such as matrix multiplication. This paper presents an investigation into the design and implementation of different matrix algorithms such as matrix operations, matrix transforms and matrix decompositions using a novel custom coprocessor system for MATrix algorithms based on Reconfigurable Computing (RCMAT). The proposed RCMAT architectures are scalable, modular and require less area and time complexity with reduced latency when compared with existing structures.

RELATIONSHIP BETWEEN WORD AND BIT LEVEL SYSTOLIC ARRAYS AS APPLIED TO MATRIX multiplied by MATRIX MULTIPLICATIONS.

The mapping of matrix multiplied by matrix multiplication onto both word and bit level systolic arrays has been investigated. It has been found that well defined word and bit level data flow constraints must be satisfied within such circuits. An efficient and highly regular bit level array has been generated by exploiting the basic compatibilities in data flow symmetries at each level of the problem. A description of the circuit which emerges is given and some details relating to its practical implementation are discussed.