{"title":"Lower bounds for matrix product","authors":"Amir Shpilka","doi":"10.1109/SFCS.2001.959910","DOIUrl":null,"url":null,"abstract":"We prove lower bounds on the number of product gates in bilinear and quadratic circuits that compute the product of two n /spl times/ n matrices over finite fields. In particular we obtain the following results: 1. We show that the number of product gates in any bilinear (or quadratic) circuit that computes the product of two n /spl times/ n matrices over GF(2) is at least 3n/sup 2/ o(n/sup 2/). 2. We show that the number of product gates in any bilinear circuit that computes the product of two n /spl times/ n matrices over GF(p) is at least (2.5 + 1.5/p/sup 3/-1)n/sup 2/ - o(n/sup 2/). These results improve the former results of N.H. Bshouty (1997) and M. Blaser (1999) who proved lower bounds of 2.5n/sup 2/ o(n/sup 2/).","PeriodicalId":378126,"journal":{"name":"Proceedings 2001 IEEE International Conference on Cluster Computing","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2001-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"44","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings 2001 IEEE International Conference on Cluster Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/SFCS.2001.959910","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 44
Abstract
We prove lower bounds on the number of product gates in bilinear and quadratic circuits that compute the product of two n /spl times/ n matrices over finite fields. In particular we obtain the following results: 1. We show that the number of product gates in any bilinear (or quadratic) circuit that computes the product of two n /spl times/ n matrices over GF(2) is at least 3n/sup 2/ o(n/sup 2/). 2. We show that the number of product gates in any bilinear circuit that computes the product of two n /spl times/ n matrices over GF(p) is at least (2.5 + 1.5/p/sup 3/-1)n/sup 2/ - o(n/sup 2/). These results improve the former results of N.H. Bshouty (1997) and M. Blaser (1999) who proved lower bounds of 2.5n/sup 2/ o(n/sup 2/).
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