基于行列子集选择的Frobenius范数低秩逼近
Low-rank approximation in the Frobenius norm by column and row subset selection
摘要：
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。
英文摘要：
A CUR approximation of a matrix $A$ is a particular type of low-rank approximation $A \approx C U R$, where $C$ and $R$ consist of columns and rows of $A$, respectively. One way to obtain such an approximation is to apply column subset selection to $A$ and $A^T$. In this work, we describe a numerically robust and much faster variant of the column subset selection algorithm proposed by Deshpande and Rademacher, which guarantees an error close to the best approximation error in the Frobenius norm. For cross approximation, in which $U$ is required to be the inverse of a submatrix of $A$ described by the intersection of $C$ and $R$, we obtain a new algorithm with an error bound that stays within a factor $k + 1$ of the best rank-$k$ approximation error in the Frobenius norm. To the best of our knowledge, this is the first deterministic polynomial-time algorithm for which this factor is bounded by a polynomial in $k$. Our derivation and analysis of the algorithm is based on derandomizing a recent existence result by Zamarashkin and Osinsky. To illustrate the versatility of our new column subset selection algorithm, an extension to low multilinear rank approximations of tensors is provided as well.
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