Subset selection for matrices is the task of extracting a column sub-matrix from a given matrix B ∈ ℝ^n×m with m > n such that the pseudoinverse of the sampled matrix has as small Frobenius or spectral norm as possible. In this paper, we consider a more general problem of subset selection for matrices that allows a block to be fixed at the beginning. Under this setting, we provide a deterministic method for selecting a column submatrix from B. We also present a bound for both the Frobenius and spectral norms of the pseudoinverse of the sampled matrix, showing that the bound is asymptotically optimal. The main technology for proving this result is the interlacing families of polynomials developed by Marcus, Spielman and Srivastava. This idea also results in a deterministic greedy selection algorithm that produces the sub-matrix promised by our result.

矩阵的子集选择是从给定矩阵B ∈ ℝ ^n×m中提取列子矩阵的任务，其中m > n使得采样矩阵的伪逆具有尽可能小的 Frobenius 或谱范数。在本文中，我们考虑了一个更一般的矩阵子集选择问题，它允许在开始时固定一个块。在此设置下，我们提供了一种从B 中选择列子矩阵的确定性方法. 我们还给出了采样矩阵伪逆的 Frobenius 和谱范数的界限，表明该界限是渐近最优的。证明这一结果的主要技术是由 Marcus、Spielman 和 Srivastava 开发的交错多项式族。这个想法也导致了一个确定性的贪婪选择算法，它产生了我们的结果所承诺的子矩阵。