Optimal $\ell_1$ Column Subset Selection and a Fast PTAS for Low Rank Approximation

We study the problem of entrywise $\ell1$ low rank approximation. We give the first polynomial time column subset selection-based $\ell1$ low rank approximation algorithm sampling $\tilde{O}(k)$ columns and achieving an $\tilde{O}(k^{{1/2})$-approximation} for any $k$, improving upon the previous best $\tilde{O}(k)$-approximation and matching a prior lower bound for column subset selection-based $\ell1$-low rank approximation which holds for any $\text{poly}(k)$ number of columns. We extend our results to obtain tight upper and lower bounds for column subset selection-based $\ellp$ low rank approximation for any $1 < p < 2$, closing a long line of work on this problem. We next give a $(1 + \varepsilon)$-approximation algorithm for entrywise $\ellp$ low rank approximation, for $1 \leq p < 2$, that is not a column subset selection algorithm. First, we obtain an algorithm which, given a matrix $A \in \mathbb{R}^{n \times d}$, returns a rank-$k$ matrix $\hat{A}$ in $2^{{\text{poly}(k/\varepsilon)}} + \text{poly}(nd)$ running time such that: $$|A - \hat{A}|p \leq (1 + \varepsilon) \cdot OPT + \frac{\varepsilon}{\text{poly}(k)}|A|p$$ where $OPT = \min{Ak \text{ rank }k} |A - Ak|p$. Using this algorithm, in the same running time we give an algorithm which obtains error at most $(1 + \varepsilon) \cdot OPT$ and outputs a matrix of rank at most $3k$ -- these algorithms significantly improve upon all previous $(1 + \varepsilon)$- and $O(1)$-approximation algorithms for the $\ellp$ low rank approximation problem, which required at least $n^{{\text{poly}(k/\varepsilon)}$} or $n^{{\text{poly}(k)}$} running time, and either required strong bit complexity assumptions (our algorithms do not) or had bicriteria rank $3k$. Finally, we show hardness results which nearly match our $2^{{\text{poly}(k)}} + \text{poly}(nd)$ running time and the above additive error guarantee.