An Improved Approximation Algorithm for the Column Subset Selection Problem

Details An Improved Approximation Algorithm for the Column Subset Selection Problem An Improved Approximation Algorithm for the Column Subset Selection Problem

2008-12-22

arxiv.org/abs/0812.4293v2

Computer Science

Data Structures and Algorithms

17 pages; corrected a bug in the spectral norm bound of the previous version

Scientific paper

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 $\Theta(k \log k)$ 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, $$ \FNorm{A - P_CA} \leq \Theta(k \log^{1/2} k) \FNorm{A-A_k}. $$ 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, $$ \TNorm{A - P_CA} \leq \Theta(k \log^{1/2} k)\TNorm{A-A_k} + \Theta(k^{3/4}\log^{1/4}k)\FNorm{A-A_k}. $$ 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.

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