An Efficient Solution to Structured Optimization Problems using Recursive Matrices
We present a linear algebra framework for structured matrices and general optimization problems. The matrices and matrix operations are defined recursively to efficiently capture complex structures and enable advanced compiler optimization. In addition to common dense and sparse matrix types, we def...
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| Published in: | Computer graphics forum Vol. 38; no. 8; pp. 33 - 39 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Oxford
Blackwell Publishing Ltd
01.11.2019
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| ISSN: | 0167-7055, 1467-8659 |
| Online Access: | Get full text |
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| Abstract | We present a linear algebra framework for structured matrices and general optimization problems. The matrices and matrix operations are defined recursively to efficiently capture complex structures and enable advanced compiler optimization. In addition to common dense and sparse matrix types, we define mixed matrices, which allow every element to be of a different type. Using mixed matrices, the low‐ and high‐level structure of complex optimization problems can be encoded in a single type. This type is then analyzed at compile time by a recursive linear solver that picks the optimal algorithm for the given problem. For common computer vision problems, our system yields a speedup of 3–5 compared to other optimization frameworks. The BLAS performance is benchmarked against the MKL library. We achieve a significant speedup in block‐SPMV and block‐SPMM. This work is implemented and released open‐source as a header‐only extension to the C+ + math library Eigen. |
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| AbstractList | We present a linear algebra framework for structured matrices and general optimization problems. The matrices and matrix operations are defined recursively to efficiently capture complex structures and enable advanced compiler optimization. In addition to common dense and sparse matrix types, we define mixed matrices, which allow every element to be of a different type. Using mixed matrices, the low‐ and high‐level structure of complex optimization problems can be encoded in a single type. This type is then analyzed at compile time by a recursive linear solver that picks the optimal algorithm for the given problem. For common computer vision problems, our system yields a speedup of 3–5 compared to other optimization frameworks. The BLAS performance is benchmarked against the MKL library. We achieve a significant speedup in block‐SPMV and block‐SPMM. This work is implemented and released open‐source as a header‐only extension to the C+ + math library Eigen. |
| Author | Stamminger, M. Rückert, D. |
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| Cites_doi | 10.1145/2866569 10.1109/MCSE.2009.207 10.1109/IPDPS.2013.80 10.1137/S0895479894278952 10.1109/SC.2016.58 10.1016/0024-3795(86)90159-X 10.1145/2892632 10.1137/0914063 10.1145/3132188 10.1145/383738.383741 10.1109/ICPP.2008.45 10.1109/IROS.2012.6385773 10.1109/MITS.2010.939925 10.1137/S0036144503428693 10.1007/978-3-642-15552-9_3 10.1145/1391989.1391995 10.1007/11557654_91 10.1007/BF02165411 10.1007/BFb0067700 10.1145/1362622.1362674 10.1145/1377603.1377607 |
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| Copyright | 2019 The Author(s) Computer Graphics Forum © 2019 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. 2019 The Eurographics Association and John Wiley & Sons Ltd. |
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| SubjectTerms | Algorithms CCS Concepts Computer vision Computing methodologies → Symbolic and algebraic algorithms Linear algebra Linear algebra algorithms Mathematical analysis Matrix methods Optimization Optimization algorithms Sparse matrices Structured matrices |
| Title | An Efficient Solution to Structured Optimization Problems using Recursive Matrices |
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