In several applications, the data consists of an ¡£¢¥ ¤ matrix ¦ and it is of interest to find an § approximation of a specified ¨ rank to ¦ ¨ where, is much smaller than ¡ and ¤. Traditional methods like the Singular Value Decomposition (SVD) help us find the “best ” such approximation. However, these methods take time polynomial in ¡�©� ¤ which is often too prohibitive. In this paper, we develop an algorithm which is qualitatively faster provided we may sample the entries of the matrix according to a natural probability distribution. Indeed, in the applications such sampling is possible. Our main result is that we can find the description of a §� � matrix of rank at ¨ most so tha
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