# A PTAS for $\ell_p$-Low Rank Approximation

@inproceedings{Ban2018APF, title={A PTAS for \$\ell_p\$-Low Rank Approximation}, author={Frank Ban and Vijay Bhattiprolu and Karl Bringmann and Pavel Kolev and Euiwoong Lee and David P. Woodruff}, booktitle={SODA 2019}, year={2018} }

A number of recent works have studied algorithms for entrywise $\ell_p$-low rank approximation, namely algorithms which given an $n \times d$ matrix $A$ (with $n \geq d$), output a rank-$k$ matrix $B$ minimizing $\|A-B\|_p^p=\sum_{i,j} |A_{i,j} - B_{i,j}|^p$. We show the following:
On the algorithmic side, for $p \in (0,2)$, we give the first $n^{\text{poly}(k/\epsilon)}$ time $(1+\epsilon)$-approximation algorithm. For $p = 0$, there are various problem formulations, a common one being the… CONTINUE READING

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