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• W2982489076 abstract "We study the low rank approximation problem of any given matrix $A$ over $mathbb{R}^{ntimes m}$ and $mathbb{C}^{ntimes m}$ in entry-wise $ell_p$ loss, that is, finding a rank-$k$ matrix $X$ such that $|A-X|_p$ is minimized. Unlike the traditional $ell_2$ setting, this particular variant is NP-Hard. We show that the algorithm of column subset selection, which was an algorithmic foundation of many existing algorithms, enjoys approximation ratio $(k+1)^{1/p}$ for $1le ple 2$ and $(k+1)^{1-1/p}$ for $pge 2$. This improves upon the previous $O(k+1)$ bound for $pge 1$ cite{chierichetti2017algorithms}. We complement our analysis with lower bounds; these bounds match our upper bounds up to constant $1$ when $pgeq 2$. At the core of our techniques is an application of emph{Riesz-Thorin interpolation theorem} from harmonic analysis, which might be of independent interest to other algorithmic designs and analysis more broadly. As a consequence of our analysis, we provide better approximation guarantees for several other algorithms with various time complexity. For example, to make the algorithm of column subset selection computationally efficient, we analyze a polynomial time bi-criteria algorithm which selects $O(klog m)$ columns. We show that this algorithm has an approximation ratio of $O((k+1)^{1/p})$ for $1le ple 2$ and $O((k+1)^{1-1/p})$ for $pge 2$. This improves over the best-known bound with an $O(k+1)$ approximation ratio. Our bi-criteria algorithm also implies an exact-rank method in polynomial time with a slightly larger approximation ratio." @default.
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• W2982489076 date "2019-10-30" @default.
• W2982489076 modified "2023-09-27" @default.
• W2982489076 title "Optimal Analysis of Subset-Selection Based L_p Low Rank Approximation" @default.
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