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Fast Matrixvector Multiplications for Largescale Logistic Regression on Sharedmemory Systems
"... Abstract—Sharedmemory systems such as regular desktops now possess enough memory to store large data. However, the training process for data classification can still be slow if we do not fully utilize the power of multicore CPUs. Many existing works proposed parallel machine learning algorithms by ..."
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Abstract—Sharedmemory systems such as regular desktops now possess enough memory to store large data. However, the training process for data classification can still be slow if we do not fully utilize the power of multicore CPUs. Many existing works proposed parallel machine learning algorithms by modifying serial ones, but convergence analysis may be complicated. Instead, we do not modify machine learning algorithms, but consider those that can take the advantage of parallel matrix operations. We particularly investigate the use of parallel sparse matrixvector multiplications in a Newton method for largescale logistic regression. Various implementations from easy to sophisticated ones are analyzed and compared. Results indicate that under suitable settings excellent speedup can be achieved. Keywordssparse matrix; parallel matrixvector multiplication; classification; Newton method I.
Efficient Multithreaded Untransposed, Transposed or Symmetric Sparse MatrixVector Multiplication with the Recursive Sparse Blocks Format
, 2014
"... In earlier work we have introduced the “Recursive Sparse Blocks ” (RSB) sparse matrix storage scheme oriented towards cache efficient matrixvector multiplication (SpMV) and triangular solution (SpSV) on cache based shared memory parallel computers. Both the transposed (SpMV T) and symmetric (SymSpM ..."
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In earlier work we have introduced the “Recursive Sparse Blocks ” (RSB) sparse matrix storage scheme oriented towards cache efficient matrixvector multiplication (SpMV) and triangular solution (SpSV) on cache based shared memory parallel computers. Both the transposed (SpMV T) and symmetric (SymSpMV) matrixvector multiply variants are supported. RSB stands for a metaformat: it recursively partitions a rectangular sparse matrix in quadrants; leaf submatrices are stored in an appropriate traditional format — either Compressed Sparse Rows (CSR) or Coordinate (COO). In this work, we compare the performance of our RSB implementation of SpMV, SpMV T, SymSpMV to that of the stateoftheart Intel Math Kernel Library (MKL) CSR implementation on the recent Intel’s Sandy Bridge processor. Our results with a few dozens of real world large matrices suggest the efficiency of the approach: in all of the cases, RSB’s SymSpMV (and in most cases, SpMV T as well) took less than half of MKL CSR’s time; SpMV ’s advantage was smaller. Furthermore, RSB’s SpMV T is more scalable than MKL’s CSR, in that it performs almost as well as SpMV. Additionally, we include comparisons to the stateofthe art format Compressed Sparse Blocks (CSB) implementation. We observed RSB to be slightly superior to CSB in SpMV T, slightly inferior in SpMV, and better (in most cases by a factor of two or more) in SymSpMV. Although RSB is a nontraditional storage format and thus needs a special constructor, it can be assembled from CSR or any other similar rowordered representation arrays in the time of a few dozens of matrixvector multiply executions. Thanks to its significant advantage over MKL’s CSR routines for symmetric or transposed matrixvector multiplication, in most of the observed cases the assembly cost has been observed to amortize with fewer than fifty iterations.
TREEBASED SPACE EFFICIENT FORMATS FOR STORING THE STRUCTURE OF SPARSE MATRICES ∗
"... Sparse storage formats describe a way how sparse matrices are stored in a computer memory. Extensive research has been conducted about these formats in the context of performance optimization of the sparse matrixvector multiplication algorithms, but memory efficient formats for storing sparse matri ..."
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Sparse storage formats describe a way how sparse matrices are stored in a computer memory. Extensive research has been conducted about these formats in the context of performance optimization of the sparse matrixvector multiplication algorithms, but memory efficient formats for storing sparse matrices are still under development, since the commonly used storage formats (like COO or CSR) are not sufficient. In this paper, we propose and evaluate new storage formats for sparse matrices that minimize the space complexity of information about matrix structure. The first one is based on arithmetic coding and the second one is based on binary tree format. We compare the space complexity of common storage formats and our new formats and prove that the latter are considerably more space efficient. Key words: sparse matrix representation; parallel execution; space efficiency; arithmeticalcodingbased format; minimal binary tree format; minimal quadtree format; AMS subject classifications. 68M14, 68W10, 68P05, 68P20, 94A17