The problem of column subset selection asks for a subset of columns from an input matrix such that the matrix can be reconstructed as accurately as possible within the span of the selected columns. A natural extension is to consider a setting where the matrix rows are partitioned into two groups, and the goal is to choose a subset of columns that minimizes the maximum reconstruction error of both groups, relative to their respective best rank-k approximation. Extending the known results of column subset selection to this fair setting is not straightforward: in certain scenarios it is unavoidable to choose columns separately for each group, resulting in double the expected column count. We propose a deterministic leverage-score sampling strategy for the fair setting and show that sampling a column subset of minimum size becomes NP-hard in the presence of two groups. Despite these negative results, we give an approximation algorithm that guarantees a solution within 1.5 times the optimal solution size. We also present practical heuristic algorithms based on rank-revealing QR factorization. Finally, we validate our methods through an extensive set of experiments using real-world data.

翻译：列子集选择问题要求从输入矩阵中选出一个列子集，使得矩阵在所选列张成的空间内能够被尽可能精确地重构。一个自然的扩展是考虑矩阵行被划分为两组的情形，其目标在于选择一个列子集，使得两组数据相对于各自最佳秩k近似的重构误差最大值最小化。将列子集选择的已知结果推广到这一公平设定并非易事：在某些场景下，不可避免地需要为每组分别选择列，从而导致预期列数翻倍。我们针对公平设定提出了一种确定性杠杆值采样策略，并证明了当存在两组数据时，采样最小规模的列子集是NP难问题。尽管存在这些负面结果，我们给出了一种近似算法，能够保证解在最优解规模的1.5倍以内。我们还提出了基于秩揭示QR分解的实用启发式算法。最后，我们通过使用真实世界数据的大量实验验证了所提方法的有效性。