  We consider the problem of matrix column subset selection, which selects a
subset of columns from an input matrix such that the input can be well
approximated by the span of the selected columns. Column subset selection has
been applied to numerous real-world data applications such as population
genetics summarization, electronic circuits testing and recommendation systems.
In many applications the complete data matrix is unavailable and one needs to
select representative columns by inspecting only a small portion of the input
matrix. In this paper we propose the first provably correct column subset
selection algorithms for partially observed data matrices. Our proposed
algorithms exhibit different merits and limitations in terms of statistical
accuracy, computational efficiency, sample complexity and sampling schemes,
which provides a nice exploration of the tradeoff between these desired
properties for column subset selection. The proposed methods employ the idea of
feedback driven sampling and are inspired by several sampling schemes
previously introduced for low-rank matrix approximation tasks (Drineas et al.,
2008; Frieze et al., 2004; Deshpande and Vempala, 2006; Krishnamurthy and
Singh, 2014). Our analysis shows that, under the assumption that the input data
matrix has incoherent rows but possibly coherent columns, all algorithms
provably converge to the best low-rank approximation of the original data as
number of selected columns increases. Furthermore, two of the proposed
algorithms enjoy a relative error bound, which is preferred for column subset
selection and matrix approximation purposes. We also demonstrate through both
theoretical and empirical analysis the power of feedback driven sampling
compared to uniform random sampling on input matrices with highly correlated
columns.
