We give the first single-pass streaming algorithm for Column Subset Selection with respect to the entrywise $\ell_p$-norm with $1 \leq p < 2$. We study the $\ell_p$ norm loss since it is often considered more robust to noise than the standard Frobenius norm. Given an input matrix $A \in \mathbb{R}^{d \times n}$ ($n \gg d$), our algorithm achieves a multiplicative $k^{\frac{1}{p} - \frac{1}{2}}\text{poly}(\log nd)$-approximation to the error with respect to the best possible column subset of size $k$. Furthermore, the space complexity of the streaming algorithm is optimal up to a logarithmic factor. Our streaming algorithm also extends naturally to a 1-round distributed protocol with nearly optimal communication cost. A key ingredient in our algorithms is a reduction to column subset selection in the $\ell_{p,2}$-norm, which corresponds to the $p$-norm of the vector of Euclidean norms of each of the columns of $A$. This enables us to leverage strong coreset constructions for the Euclidean norm, which previously had not been applied in this context. We also give the first provable guarantees for greedy column subset selection in the $\ell_{1, 2}$ norm, which can be used as an alternative, practical subroutine in our algorithms. Finally, we show that our algorithms give significant practical advantages on real-world data analysis tasks.
翻译:我们给出了列子选择首个单流流算法, 以 1 美元\ ell_ p$- norm 和 1 美元\ leq p < 2 美元 。 我们研究$ ell_ p$ 常规损失, 因为它通常被认为比标准的 Frobenius 规范对噪音更强。 鉴于输入矩阵 $A\ in \ mathbb{R ⁇ d\ times n} $ (美元\ gg d$ ), 我们的算法实现了一个倍增 $k ⁇ frac{ 1\\ { } -\ frac{ text{ 2}\ text{ { polly} (\ log nd) 。 我们的算法中的一种关键元素是 $_ laxxxxxxxxxxxx