Simple and practical algorithms for ℓ_p-norm low-rank approximation

05/24/2018

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by   Anastasios Kyrillidis, et al.

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We propose practical algorithms for entrywise ℓ_p-norm low-rank approximation, for p = 1 or p = ∞. The proposed framework, which is non-convex and gradient-based, is easy to implement and typically attains better approximations, faster, than state of the art. From a theoretical standpoint, we show that the proposed scheme can attain (1 + ε)-OPT approximations. Our algorithms are not hyperparameter-free: they achieve the desiderata only assuming algorithm's hyperparameters are known a priori---or are at least approximable. I.e., our theory indicates what problem quantities need to be known, in order to get a good solution within polynomial time, and does not contradict to recent inapproximabilty results, as in [46].