Simple and practical algorithms for $\ell_p$-norm low-rank approximation
This work addresses a specific computational challenge in low-rank approximation for machine learning and data analysis, offering practical improvements but is incremental in nature.
The paper tackles the problem of entrywise ℓ_p-norm low-rank approximation for p=1 or p=∞ by proposing a non-convex, gradient-based algorithm that is easy to implement and typically achieves better approximations faster than state-of-the-art methods, with theoretical guarantees of (1+ε)-OPT approximations.
We propose practical algorithms for entrywise $\ell_p$-norm low-rank approximation, for $p = 1$ or $p = \infty$. 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 $(1 + \varepsilon)$-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].