sparse matrix benchmarksparse matrix benchmark

The Sparse Matrix-Vector Multiplication (SpMV) kernel ranks among the most important and thoroughly studied linear algebra operations, as it lies at the heart of many iterative methods for the solution of sparse linear systems, and often constitutes a severe performance bottleneck. ACM Trans. (TPDS) 24, 10 (2013), 1930--1940. Syst. 2 Answers Sorted by: 6 Update June 2021: The missing specialized algorithm for sparse views mentioned below is now implemented, so the performance is much more reasonable these days (Julia 1.6+): julia> @btime A*v; 2.063 s (4 allocations: 23.84 KiB) julia> @btime B*v; 2.836 s (9 allocations: 25.30 KiB) The ACM Digital Library is published by the Association for Computing Machinery. 36, 5 (2014), 401--423. Technical Report NISTIR-5935. Intel Math Kernel Library. In this case the CSR representation contains 13 entries, compared to 16 in the original matrix. 16, 521 (2005). ITRS. ACM, New York, Article 50. 2009. Scalable Parallel Generation of Very Large Sparse Benchmark Matrices See the about page for more information. See the object documentation for the top-level benchmark functions and the microbenchmark definition classes listed below for information on how to configure the individual microbenchmarks. Chapman & Hall, Boca Raton (1997), Carney, S.: A revised proposal for a sparse blas toolkit (1994), Dongarra, J., Lumsdaine, A., Pozo, R., Remington, K.: A sparse matrix library in c++ for high performance architectures (1994), Dongarra, J.J., Van der Vorst, H.A. USENIX, San Jose (2012). See file LICENSE.txt for details. This operation appears when forming the normal equations of interior point methods It is likely known as the Yale format because it was proposed in the 1977 Yale Sparse Matrix Package report from Department of Computer Science at Yale University.[11]. Cluster Computing Unable to display preview. Lecture Notes in Computer Science, vol 2763. It also appears, , where matrix Unlike in the case of dense matrices, handling them does not entail much reuse of data. IEEE. In fact, many of the linear algebra applications that benefit from sparsity have over 99% sparsity in their matrices. See the object documentation for the RunMachineLearningBenchmark, pam, and clara functions for more details. This format allows fast row access and matrix-vector multiplications (Mx). Jeremiah Willcock and Andrew Lumsdaine. Software 5, 1 (1979), 18--35. The challenge arises because of the following reasons: (i) Owing to the limitation of the analog computing mechanism, specifically Ohm's law and Kirchhoff's law, matrix elements must be mapped to array cells corresponding to their positions in the matrix for correct calculation. PDF Model-guided Performance Analysis of the Sparse Matrix-Matrix PVLDB 4(11), 11111122 (2011), Huss-Lederman, S., Jacobson, E.M., Johnson, J.R., Tsao, A., Turnbull, T.: Implementation of Strassens algorithm for matrix multiplication. This is another format that is good for incremental matrix construction. Exp. Scalable Parallel Generation of Very Large Sparse Benchmark Matrices. An overview of the Trilinos project. USENIX Association, Berkeley, p. 10 (2004). http://people.sc.fsu.edu/~jburkardt/pdf/hbsmc.pdf (1992). M. A. Heroux, D. W. Doerfler, P. S. Crozier, J. M. Willenbring, H. C. Edwards, A. Willians, M. Rajan, E. R. Keiter, H. K. Thornquist, and R. W. Numrich. The sparse matrix microbenchmarks supported by the sparse matrix benchmark are: matrix-vector multiplication, Cholesky factorization, LU factorization, and QR factorization. ACM, New York, 157--172. 77(4), 802813 (2008), Foldi, T., von Csefalvay, C., Perez, N.A. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. Big Data (2020). (eds.) The name is based on the fact that column index information is compressed relative to the COO format. The right array stores nonzero values in consecutive blocks, while the second array contains the column indices of the corresponding nonzero blocks. The Collection is widely used by the numerical linear algebra community for the development and performance evaluation of sparse matrix algorithms. http://dl.acm.org/citation.cfm?id=2999325.2999464, Son, M., Lee, K.: Distributed matrix multiplication performance estimator for machine learning jobs in cloud computing. The clustering benchmark generates clusters of normally distributed feature vectors in an \(N\)-dimensional, real-valued feature space where the mean of one cluster is located at the origin and the means of two clusters are each located at positions -1 and 1 of one or more of the axes. Orthogonalization methods (such as QR factorization) are common, for example, when solving problems by least squares methods. The function returns a vector of ClusteringMicrobenchmark objects specifying each microbenchmark. Springer International Publishing, Cham, 24--45. By contrast, if most of the elements are non-zero, the matrix is considered dense. Springer, Berlin, 127--136. Results obtained in the comparison of the SparseX library with two other libraries are discussed. https://www.usenix.org/conference/nsdi12/technical-sessions/presentation/wieder, Yadwadkar, N.J., Hariharan, B., Gonzalez, J.E., Smith, B., Katz, R.H.: Selecting the best VM across multiple public clouds: a data-driven performance modeling approach. These occur frequently in linear systems generated by the nite element method (FEM), for example, and are naturally suited for register blocking optimizations. IEEE Trans. Archit. Ann. In Proceedings of the ISCA 25th International Conference on Computers and Their Applications (CATA10). The online phase is for partitioning the input sparse matrix and computing the execution plan. A benchmark of sparse matrix dense vector multiplication in C++ using homebuilt and pre-packaged methods. 2014. Each microbenchmark is tested with several matrices of increasing size. Y. Saad. Benchmark for TVM: Currently, clustering microbenchmarks are the only microbenchmarks supported by the machine learning benchmark. You signed in with another tab or window. We offer the collection to other researchers as a standard benchmark for comparative studies of algorithms. The university of Florida sparse matrix collection | ACM Transactions Software 38 (2011), 1--25. http://www.cise.ufl.edu/research/sparse/matrices. cuSPARSE | NVIDIA Developer STREAM: Sustainable Memory Bandwidth in High Performance Computing. {\displaystyle Ax_{i}} cuSPARSE is widely used by engineers and scientists working on applications such as machine learning, computational fluid dynamics, seismic exploration and computational sciences. D. H. Bailey, E. Barscz, J. T. Barton, D. S. Browning, R. L. Carter, L. Dagum, R. A. Fatoohi, P. O. Frederickson, T. A. Lasinksi, R. S. Schreiber, H. D. Simon, V. Venkatakrishnan, and S. K. Weeratunga. Performance evaluation of the sparse matrix-vector multiplication on modern architectures. x The Sparse Matrix Benchmark Next:Test problemsUp:Comparing the performance ofPrevious:The Test Platforms The Sparse Matrix Benchmark In our first study the benchmark we choose is a simple yet important operation: that of multiplying two sparse matrices. Accessed 27 March 2011, Hernandez, V., Roman, J.E., Vidal, V.: SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. In the case of a sparse matrix, substantial memory requirement reductions can be realized by storing only the non-zero entries. Institute of Computer and Information Science, Czestochowa University of Technology, Czestochowa, Poland, University of Tennessee, Department of Computer Science, Knoxville, Tennessee, USA, Technical University of Denmark Informatics and Mathematical Modelling, Kongens Lyngby, Denmark, Langr, D., imeek, I., Tvrdk, P., Dytrych, T. (2014). large scale unstructured calculations where a multilevel/multigrid 1999. The latest version of cuSPARSE can be found in the CUDA Toolkit. For better performance, it is important to satisfy the following conditions: For this new storage format, perform similar steps as with CSR and COO cusparseSpMM. Jiajia Li, Guangming Tan, Mingyu Chen, and Ninghui Sun. NVIDIA cuSPARSELt is a high-performance CUDA library dedicated to general matrix-matrix operations in which at least one operand is a sparse matrix: where refers to in-place operations such as transpose/non-transpose, and are scalars. Numerical Methods for Large Eigenvalue Problems. Archit. The focus is on providing a benchmark suite which is flexible and easy to port on (novel) systems, yet complete enough to expose the main difficulties which are encountered when dealing with sparse matrices. By contrast, if the same line of balls were to have springs connecting each ball to all other balls, the system would correspond to a dense matrix. These keywords were added by machine and not by the authors. A high performance algorithm using pre-processing for the sparse matrix-vector multiplication. University of California at Berkeley, Berkeley, CA. To manage your alert preferences, click on the button below. However, if the functionality being microbenchmarked is implemented with support for multithreading, and the number of threads can be controlled through the use of environment variables, as is often the case, then the benchmarks can be executed multithreaded. You switched accounts on another tab or window. SIAM. 38(1), 1:11:25 (2011), MathSciNet 2010, 721726 (2010), Shahidinejad, A., Ghobaei-Arani, M., Masdari, M.: Resource provisioning using workload clustering in cloud computing environment: a hybrid approach. A tag already exists with the provided branch name. Ali Pinar and Michael T. Heath. 1979. Even though the benchmark functions do not control the number of threads to be utilized, the benchmarks must still report the number of threads used in the CSV files and data frames generated for reporting results. R HPC Benchmark - The Comprehensive R Archive Network About the benchmark SparseBench is a benchmark suite of iterative methods on sparse data. Basic linear algebra subprograms for Fortran usage. The pam function implements the partitioning around medoids algorithm which has quadratic time complexity. ACM, New York, 339--350. https://doi.org/10.1214/aos/1013203451, Article A discussion of three storage formats for sparse matrices follows: a) the compressed sparse row (CSR) format, b) the blocked compressed sparse row (BCSR) format, and c) the CSX format. 23272336 (2015), Herodotou, H., Babu, S.: Profiling, what-if analysis, and cost-based optimization of mapreduce programs. All of the dense linear algebra kernels are implemented around BLAS or LAPACK interfaces. The benchmark algorithms (operations) are categorized in (a) value related operations and (b) position related operations. Cache-aware roofline model: Upgrading the loft. Use 128-byte aligned pointers for matrices for vectorized memory access. The novelty compared to previous benchmarks is that it is not limited by the need for a compiler. Optimizing sparse matrix-vector multiplication using index and value compression. 17(1), 12351241 (2016), MathSciNet The paper then presents a performance analysis for the sparse matrix-vector multiplication for each of these three storage formats. J. D. McCalpin. Let NNZ denote the number of nonzero entries in M. (Note that zero-based indices shall be used here.). 2003. The following table lists the microbenchmarks supported by the sparse matrix benchmark, the matrices each microbenchmark is executed with, and the kernel function that is microbenchmarked. https://doi.org/10.1007/978-3-642-55224-3_18, DOI: https://doi.org/10.1007/978-3-642-55224-3_18, Publisher Name: Springer, Berlin, Heidelberg, eBook Packages: Computer ScienceComputer Science (R0). , is derived from the matrix on the fine grid, , Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. The allocator must return a list of allocated data objects, including the matrix, for the microbenchmark to operate on. IEEE Computer Society Press, 32--41. Please download or close your previous search result export first before starting a new bulk export. Retrieved from http://www.cs.virginia.edu/stream/. Be warned: these benchmarks are very special- ized on a neural network like algorithm I had to implement. Sparse matrix multiplication in a record-low power self-rectifying J. Comput. PDF Benchmarking Sparse Matrix-Vector Multiply in Five Minutes The lsfit microbenchmark tests matrices of dimension \(2N\)-by-\(N/2\), while the remaining microbenchmarks all test \(N\)-by-\(N\) (square) matrices. A unified sparse matrix data format for efficient general sparse matrix-vector multiply on modern processors with wide SIMD units. As the usual dense GEMM, the computation partitions the output matrix into tiles. Execution environments of distributed SPMM tasks on cloud resources can be set up in diverse ways with respect to the input sparse datasets, distinct SPMM implementation methods, and the choice of cloud instance types. One typically uses another format (LIL, DOK, COO) for construction. 29512959 (2012). USENIX Association, Boston, pp. pp This is a preview of subscription content, access via your institution. The use of preconditioners can significantly accelerate convergence of such iterative methods. In Proceedings of the 20th Annual International Conference on Supercomputing (ICS06). An important special type of sparse matrices is band matrix, defined as follows. One typically constructs a matrix in this format and then converts to another more efficient format for processing. National Technical University of Athens, Athens, Greece, Swiss National Supercomputing Centre, ETH Zurich, Lugano, Switzerland, IBM Reasearch Zurich, Zurich, Switzerland. Note that for the Xeon Phi KNC you have to cross-compile on the host and run the script on the device. https://www.usenix.org/conference/nsdi17/technical-sessions/presentation/alipourfard, Bosagh Zadeh, R., Meng, X., Ulanov, A., Yavuz, B., Pu, L., Venkataraman, S., Sparks, E., Staple, A., Zaharia, M.: Matrix Computations and Optimization in Apache Spark, Ser. 125137. Benchmarking is a way to assess the performance of software on a computing platform and to compare performance between different platforms. 759773 (2018). Section 4 ends with the incorporation of the kernels in some solvers for systems of linear algebraic equations based on the use of the conjugate gradient method. We acknowledge the Louisiana Optical Network Initiative (LONI) for providing HPC resources. Sandia National Laboratories. The dense matrix linear algebra kernels, sparse matrix linear algebra kernels, and machine learning functions that are benchmarked are all part of the R interpreters intrinsic functionality or packages included the with the R programming environment standard distributions from CRAN. Optimizing Sparse Matrix Computations for Register Reuse in SPARSITY. Intel Coorporation. IEEE Comput. Mach. They discuss both the application of a high-level implementation, which requires minimal user efforts, and the use of a low-level version for advanced users. The tested matrix dimensions are parameterized by \(N\) with values of \(N\) equal to: 1000, 2000, 4000, 8000, 10000, 15000, and 20000. P202/12/2011, by the U.S. National Science Foundation under Grant No. This example runs all of the default sparse matrix microbenchmarks, saves the summary statistics for each microbenchmark in the directory SparseMatrixResults, and saves the data frame returned from the dense matrix benchmark to a file named allResultsFrame.RData. 1992. Richard W. Vuduc and Hyun-Jin Moon. ACM, New York, Article 30. I am especially interested in sparse matrix multiplication for single- and multi-core systems? 55, 11 (1967), 1801--1809. Weifeng Liu and Brian Vinter. Two levels of application performance implementation are available in their library. http://hadoop.apache.org/, Friedman, J.H. The old Yale format works exactly as described above, with three arrays; the new format combines ROW_INDEX and COL_INDEX into a single array and handles the diagonal of the matrix separately.[10]. Figure 1 shows the general matrix multiplication (GEMM) operation by using the block sparse format. Improving Performance via Mini-applications. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. Parallel Distrib. If any of the microbenchmarks fails to run in a timely manner or fails due to memory constraints, the matrix sizes and number of performance trials per matrix can be adjusted. Considering the characters and hardware specifications on the cloud, we propose unique features to build a GB-regressor model and Bayesian optimizations. The user may also specify one or more warm-up runs to ensure the R programming environment is settled before executing the performance trials. 2010. A ACM, New York, 87--96. Symposium on the Birth of Numerical Analysis. https://doi.org/10.1145/263580.263591, Leskovec, J., Krevl, A.: SNAP datasets: Stanford large network dataset collection. Price excludes VAT (USA) http://arxiv.org/abs/1104.4874. High Performance Computing, pp. Correspondence to Block sizes: 32 and 16. Large sparse matrices often appear in scientific or engineering applications when solving partial differential equations. Then we make the slices V[1:2] = [8] and COL_INDEX[1:2] = [1]. ACM Trans. The authors stress that the kernels (based on sparse matrix-vector multiplication) and the use of iterative methods in large linear systems are becoming more and more popular (in comparison with the application of direct methods). The allocator returns a list of data objects, including the matrix, for the microbenchmark to operate on. USENIX Association, Boston, July 2018, pp. National Institute of Standards and Technology. The integer index is unused by the microbenchmarks specified by the GetSparse* default functions because the sparse matrix microbenchmarks read the test matrices from files as opposed to dynamically generating them. Learn. performance - Julia: view of sparse matrix - Stack Overflow 261272 (2013). The order of the matrices and the length of the vectors in these tasks are in most cases very large, but the matrices are sparse (that is, many of their elements are equal to zero) and it is necessary to exploit this property. Inexact solves in Krylov-based model reduction. Lett. C. L. Lawson, R. J. Hanson, D. R. Kincaid, and F. T. Korgh. For consistent behavior, the user should set the environment variable for the number of threads before the R programming environment is initialized. : Algorithms for non-negative matrix factorization. PubMedGoogle Scholar, Novosibirsk State Technical University, Russia, Stathis, P., Vassiliadis, S., Cotofana, S. (2003). Check if you have access through your login credentials or your institution to get full access on this article. This alert has been successfully added and will be sent to: You will be notified whenever a record that you have chosen has been cited. The benchmark performance results can also be used to prioritize software performance optimization efforts on emerging High Performance Computing (HPC) systems. A matrix is typically stored as a two-dimensional array. Efficiently processing sparse matrices is a central and performance-critical part of many scientific simulation codes. The International Journal of Supercomputer Applications5(3), 6373 (1991), CrossRef Each entry in the array represents an element ai,j of the matrix and is accessed by the two indices i and j. Conventionally, i is the row index, numbered from top to bottom, and j is the column index, numbered from left to right. The microbenchmarks, their associated identifiers and brief descriptions of the tested matrices are given in the table below. In: Proceedings of Fourth Symposium on the Frontiers of Massively Parallel Computation, vol. [4], LIL stores one list per row, with each entry containing the column index and the value. 24, 124 (2021), Shen, C., Tong, W., Choo, K.-K.R., Kausar, S.: Performance prediction of parallel computing models to analyze cloud-based big data applications. University of Florida Sparse Matrix Collection, Transpose, reshape, and retranspose matrix, eigen(A, symmetric=FALSE, only.values=FALSE), Laplacian 7-point stencil applied to 100x100x100 grid, Laplacian 7-point stencil applied to 200x200x200 grid, Undirected weighted graph from congressional redistricting, Eigenvalue problem from computer graphics/vision. This example shows how to specify a new clustering microbenchmark and run it. Syst. Springer-Verlag, Berlin, 807--816. In: Proceedings of the 24th International Symposium on High-Performance Parallel and Distributed Computing, ser. In Proceedings of the 5th Conference on Computing Frontiers (CF08). In: Supercomputing 96:Proceedings of the 1996 ACM/IEEE Conference on Supercomputing, pp. ACM, 156--165. I. S. Duff and J. K. Reid. Math. In: 2019 IEEE 12th International Conference on Cloud Computing (CLOUD), July 2019, pp. ICS 97. A number of algorithms are designed for bandwidth minimization. Software 5, 3 (1979), 308--323. DOE-0904874. IEEE, 3405--3411. https://doi.org/10.1137/1.9780898719918, Kim, J., Lee, K.: Functionbench: a suite of workloads for serverless cloud function service. International Conference on Parallel Processing and Applied Mathematics, PPAM 2013: Parallel Processing and Applied Mathematics Efficient management of parallelism in object oriented numerical software libraries. MathSciNet 2, ser. PARALUTION project. 2008. T. Gkountouvas, V. Karakasis, K. Kourtis, G. Goumas, and N. Koziris. The allocator functions for the dense matrix microbenchmarks take a DenseMatrixMicrobenchmark object specifying the microbenchmark and an integer index indicating which of the above values of \(N\) is to be applied. Society for Industrial and Applied Mathematics, Philadelphia (2003). See the object documentation for RunMachineLearningBenchmarks and the GetClusteringDefaultMicrobenchmarks functions for more details. http://doi.acm.org/10.1145/3127479.3131614, Yu, Y., Tang, M., Aref, W.G., Malluhi, Q.M., Abbas, M.M., Ouzzani, M.: In-memory distributed matrix computation processing and optimization. For fast performance of the dense matrix kernels, it is crucial to link the R programming environment with optimized BLAS and LAPACK libraries. This routine supports CSR, Coordinate (COO), as well as the new Blocked-ELL storage formats. When storing and manipulating sparse matrices on a computer, it is beneficial and often necessary to use specialized algorithms and data structures that take advantage of the sparse structure of the matrix. for large scale numerical optimization. 1991. https://doi.org/10.1007/s10586-021-03287-3, DOI: https://doi.org/10.1007/s10586-021-03287-3. For other uses, see, Toggle Storing a sparse matrix subsection. The symbolic Cholesky decomposition can be used to calculate the worst possible fill-in before doing the actual Cholesky decomposition. The data array stores non-zero matrix elements in sequential order from top to bottom along each column, then from the left-most column to the right-most. For example, CSC is (val, row_ind, col_ptr), where val is an array of the (top-to-bottom, then left-to-right) non-zero values of the matrix; row_ind is the row indices corresponding to the values; and, col_ptr is the list of val indexes where each column starts. 1952. for large sparse linear systems, the matrix on a coarse grid,