We consider the problem of selecting the best subset of exactly $k$ columns from an $m \times n$ matrix $A$. We present and analyze a novel two-stage algorithm that runs in $O(\min\{mn^2,m^2n\})$ time and returns as output an $m \times k$ matrix $C$ consisting of exactly $k$ columns of $A$. In the first (randomized) stage, the algorithm randomly selects $Θ(k \log k)$ columns according to a judiciously-chosen probability distribution that depends on information in the top-$k$ right singular subspace of $A$. In the second (deterministic) stage, the algorithm applies a deterministic column-selection procedure to select and return exactly $k$ columns from the set of columns selected in the first stage. Let $C$ be the $m \times k$ matrix containing those $k$ columns, let $P_C$ denote the projection matrix onto the span of those columns, and let $A_k$ denote the best rank-$k$ approximation to the matrix $A$. Then, we prove that, with probability at least 0.8, $$ \FNorm{A - P_CA} \leq Θ(k \log^{1/2} k) \FNorm{A-A_k}. $$ This Frobenius norm bound is only a factor of $\sqrt{k \log k}$ worse than the best previously existing existential result and is roughly $O(\sqrt{k!})$ better than the best previous algorithmic result for the Frobenius norm version of this Column Subset Selection Problem (CSSP). We also prove that, with probability at least 0.8, $$ \TNorm{A - P_CA} \leq Θ(k \log^{1/2} k)\TNorm{A-A_k} + Θ(k^{3/4}\log^{1/4}k)\FNorm{A-A_k}. $$ This spectral norm bound is not directly comparable to the best previously existing bounds for the spectral norm version of this CSSP. Our bound depends on $\FNorm{A-A_k}$, whereas previous results depend on $\sqrt{n-k}\TNorm{A-A_k}$; if these two quantities are comparable, then our bound is asymptotically worse by a $(k \log k)^{1/4}$ factor.
Low-rank approximation in the Frobenius norm by column and row subset selection
矩阵$A$的CUR近似是特定类型的低秩近似$A \approx C U R$，其中$C$和$R$分别由$A$的列和行组成。获得这种近似的一种方法是将列子集选择应用于$A$和$A^T$。在这项工作中，我们描述了一个强大的数值和更快的变形列子集选择算法提出的Deshpande和Rademacher，这保证了误差接近Frobenius范数的最佳逼近误差。对于交叉近似，其中$U$需要是由$C$和$R$的交集描述的$A$的子矩阵的逆，我们得到一个新的算法，其误差界保持在最佳秩的因子$k + 1$内- $k$近似误差在Frobenius范数。据我们所知，这是第一个确定性的多项式时间算法，这个因素是有界的多项式在$k$。我们的推导和分析的算法是基于去随机化Zamarashkin和Osinsky最近存在的结果。为了说明我们的新的列子集选择算法的多功能性，低多线性秩张量近似的扩展。
我们计算的经典最小二乘问题$\min_x||A x -b||,$，其中$A$是一个任意矩阵的\emph{sparse}解决方案。我们描述了一种新的算法，这个稀疏最小二乘问题。该算法的操作如下：首先，它从$A$中选择列，然后仅用所选列求解最小二乘问题。我们使用的列选择算法是已知的，以及研究列子集选择问题。这篇文章的贡献是表明，它给出了有利的结果，以及稀疏最小二乘。具体来说，我们证明了我们的算法得到的解决方案向量是接近的解决方案向量通过所谓的“SVD截断正则化方法”。
On Socially Fair Low-Rank Approximation and Column Subset Selection
Simons Institute for the Theory of Computing（西蒙斯理论计算研究所）；UC Berkeley（加州大学伯克利分校）；Toyota Technological Institute at Chicago（芝加哥丰田理工学院）；Carnegie Mellon University（卡内基梅隆大学）；Google Research（谷歌研究院）；Texas A&M University（德克萨斯农工大学）
