基于列的最优低秩矩阵重构
Optimal Column-Based Low-Rank Matrix Reconstruction
摘要：
我们证明了，对于任何实值矩阵$X \in \R^{m \times n}$，正整数$r \ge k$，有一个子集的$r$列$X$，使投影$X$到他们的跨度给出了一个$\sqrt{\frac{r+1}{r-k+1}}$ -近似的最佳秩- $k$近似的$X$在Frobenius范数。我们表明，我们实现的列数和近似比之间的权衡是最佳的低阶项。此外，存在一种确定性算法来找到在$O(r n m^ω \log m)$算术运算中运行的列的这样的子集，其中$ω$是矩阵乘法的指数。我们还给出了一个更快的随机算法，运行在$O(r n m^2)$算术运算。
英文摘要：
We prove that for any real-valued matrix $X \in \R^{m \times n}$, and positive integers $r \ge k$, there is a subset of $r$ columns of $X$ such that projecting $X$ onto their span gives a $\sqrt{\frac{r+1}{r-k+1}}$-approximation to best rank-$k$ approximation of $X$ in Frobenius norm. We show that the trade-off we achieve between the number of columns and the approximation ratio is optimal up to lower order terms. Furthermore, there is a deterministic algorithm to find such a subset of columns that runs in $O(r n m^ω \log m)$ arithmetic operations where $ω$ is the exponent of matrix multiplication. We also give a faster randomized algorithm that runs in $O(r n m^2)$ arithmetic operations.
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