线性时间中的低秩矩阵逼近
Low Rank Matrix Approximation in Linear Time
摘要：
给定具有$\newcommand{\MatA}{\mathcal{M}}$行和$\newcommand{\NSize}{\mathsf{N}{}}$列以及固定的$\newcommand{\Fnorm}[1]{\left\| {#1} \right\|_F}$和$\eps$的矩阵$\newcommand{\pth}[1]{\left(#1\right)}$，我们提出了一种算法，该算法在线性时间（即，$O(\NSize )$）计算具有近似误差$\Fnorm{\MatA - \MatB}^2 \leq (1+\eps) \PrcOpt{\MatA}{k}$的$k$秩矩阵$\MatB$，其中$\NSize = n d$是输入大小，而$\PrcOpt{\MatA}{k}$是$\MatA$的$k$秩近似的最小误差.该算法以恒定概率成功，据我们所知，它是第一个实现乘法近似的线性时间算法。
英文摘要：
$\newcommand{\MatA}{\mathcal{M}}$ $\newcommand{\eps}{\varepsilon}$ $\newcommand{\NSize}{\mathsf{N}{}}$ $\newcommand{\MatB}{\mathcal{B}}$ $\newcommand{\Fnorm}[1]{\left\| {#1} \right\|_F}$ $\newcommand{\PrcOpt}[2]{μ_{\mathrm{opt}}\pth{#1, #2}}$ $\newcommand{\pth}[1]{\left(#1\right)}$ Given a matrix $\MatA$ with $n$ rows and $d$ columns, and fixed $k$ and $\eps$, we present an algorithm that in linear time (i.e., $O(\NSize )$) computes a $k$-rank matrix $\MatB$ with approximation error $\Fnorm{\MatA - \MatB}^2 \leq (1+\eps) \PrcOpt{\MatA}{k}$, where $\NSize = n d$ is the input size, and $\PrcOpt{\MatA}{k}$ is the minimum error of a $k$-rank approximation to $\MatA$. This algorithm succeeds with constant probability, and to our knowledge it is the first linear-time algorithm to achieve multiplicative approximation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
