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We consider the problem of approximating a given matrix by a low-rank matrix so as to minimize the entrywise -approximation error, for any ; the case is the classical SVD problem. We obtain the first provably good approximation algorithms for this robust version of low-rank approximation that work for every value of . Our algorithms are simple, easy to implement, work well in practice, and illustrate interesting tradeoffs between the approximation quality, the running time, and the rank of the approximating matrix.