Algorithms for ℓp Low Rank Approximation

We consider the problem of approximating a given matrix by a low-rank matrix so as to minimize the entrywise ℓp-approximation error, for any p ≥ 1; the case p = 2 is the classical SVD problem. We obtain the first provably good approximation algorithms for this version of low-rank approximation that work for every value of p ≥ 1, including p = ∞. 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.