l15 - 6.896 Sublinear Time Algorithms November 02, 2004...

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Unformatted text preview: 6.896 Sublinear Time Algorithms November 02, 2004 Lecture 15 Lecturer: Eli Ben-Sasson Scribe: Swastik Kopparty 1 Approximating Matrix Product in Sublinear Time Today we consider the problem of multiplying matrices in sub-linear time. The results are from the work of Drineas, Kannan and Mahoney [1]. Given A R m n , B R n p , we wish to approximate A B R m p in sublinear time. Think of m, p n , so that the size of the output is sublinear in the size of the input. For starters, we need to define what we mean by approximating a matrix. will use the Frobenius norm as our measure of proximity. Definition 1 (Matrix Norms) For a m p matrix M , the Frobenius Norm of M is || M || F . = s X i [ m ] ,j [ p ] M 2 i,j . The L 2 norm of M is || M || 2 = sup x R n | Mx | . In our context, we will say M approximates AB if || M AB || F is small. The Frobenius norm is related to the L 2 norm by the following inequalities: || M || 2 || M || F n || M || 2 . Thus an approximation in Frobenius Norm gives some approximation in L 2 norm. We point out that [1] give more results on approximating the L 2 norm, which we wont describe. 1.1 The Sublinear Approximation Algorithm Key Observation The product AB , which is an m p matrix, is a sum of n rank one m p matrices....
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l15 - 6.896 Sublinear Time Algorithms November 02, 2004...

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