Improving compressed matrix multiplication using control variate method
The seminal work by Pagh [1] proposed a matrix multiplication algorithm for real-valued squared matrices called Compressed Matrix Multiplication (CMM) having a sparse matrix output product. The algorithm is based on a popular sketching technique called Count-Sketch [2] and Fast Fourier Transform (FF...
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| Vydáno v: | Information processing letters Ročník 187; s. 106517 |
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Elsevier B.V
01.01.2025
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| ISSN: | 0020-0190, 1872-6119 |
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| Abstract | The seminal work by Pagh [1] proposed a matrix multiplication algorithm for real-valued squared matrices called Compressed Matrix Multiplication (CMM) having a sparse matrix output product. The algorithm is based on a popular sketching technique called Count-Sketch [2] and Fast Fourier Transform (FFT). For input square matrices A and B of order n and the product matrix AB with Frobenius norm ||AB||F, the algorithm offers an unbiased estimate for each entry, i.e., (AB)i,j of the product matrix AB with a variance bounded by ||AB||F2/b, where b is the compressed bucket size. Thus, the variance will eventually become high for a small bucket size. In this work, we address the high variance problem of CMM with the help of a simple and practical technique based on classical variance reduction methods in statistics. Our techniques rely on the Control Variate (CV) method. We suggest rigorous theoretical analysis for variance reduction and complement it via supporting empirical evidence.
•This work proposes an improvement to the compressed matrix multiplication (CMM) algorithm.•Our proposal, CV-CMM, is based on the control variate (CV) method.•Our proposal provides more accurate matrix product estimates compared to CMM. |
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| AbstractList | The seminal work by Pagh [1] proposed a matrix multiplication algorithm for real-valued squared matrices called Compressed Matrix Multiplication (CMM) having a sparse matrix output product. The algorithm is based on a popular sketching technique called Count-Sketch [2] and Fast Fourier Transform (FFT). For input square matrices A and B of order n and the product matrix AB with Frobenius norm ||AB||F, the algorithm offers an unbiased estimate for each entry, i.e., (AB)i,j of the product matrix AB with a variance bounded by ||AB||F2/b, where b is the compressed bucket size. Thus, the variance will eventually become high for a small bucket size. In this work, we address the high variance problem of CMM with the help of a simple and practical technique based on classical variance reduction methods in statistics. Our techniques rely on the Control Variate (CV) method. We suggest rigorous theoretical analysis for variance reduction and complement it via supporting empirical evidence.
•This work proposes an improvement to the compressed matrix multiplication (CMM) algorithm.•Our proposal, CV-CMM, is based on the control variate (CV) method.•Our proposal provides more accurate matrix product estimates compared to CMM. |
| ArticleNumber | 106517 |
| Author | Dubey, Punit Pankaj Verma, Bhisham Dev Thakur, Manoj Pratap, Rameshwar |
| Author_xml | – sequence: 1 givenname: Bhisham Dev surname: Verma fullname: Verma, Bhisham Dev email: bhishamdevverma@gmail.com organization: Indian Institute of Technology Mandi, Himachal Pradesh, India – sequence: 2 givenname: Punit Pankaj surname: Dubey fullname: Dubey, Punit Pankaj email: punitpankajdubey@gmail.com organization: Indian Institute of Technology Mandi, Himachal Pradesh, India – sequence: 3 givenname: Rameshwar surname: Pratap fullname: Pratap, Rameshwar email: rameshwar@cse.iith.ac.in organization: Indian Institute of Technology Hyderabad, Telangana, India – sequence: 4 givenname: Manoj surname: Thakur fullname: Thakur, Manoj email: manoj@iitmandi.ac.in organization: Indian Institute of Technology Mandi, Himachal Pradesh, India |
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| Cites_doi | 10.1145/1077464.1077466 10.1137/0909040 10.1016/0022-0000(81)90033-7 10.1006/jcss.1997.1545 10.1145/2220357.2220361 10.1007/BF02165411 10.1145/2493252.2493254 10.1016/0022-0000(79)90044-8 10.1145/3019134 10.1016/S0304-3975(03)00400-6 10.1137/15M1009718 10.1006/jagm.1998.0989 10.1137/S0097539704442684 10.1287/mnsc.27.3.322 10.1007/s10994-022-06166-z 10.1007/s10618-021-00764-6 |
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| Keywords | Control variate Sketching algorithms Matrix multiplication Count-Sketch |
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