AI 中文总结
研究从实矩阵\(X\)中选\(k\)列使其体积最大的问题，证明体积对数是次模函数，对比带列主元的Businger - Golub QR和Gu - Eisenstat QR算法，发现前者误差界更好，该结论还扩展到对称正定矩阵找最大体积子矩阵。
AI 中文摘要
问题是从实\(m\times n\)矩阵\(X\)中选择\(k\)列使其体积最大。我们证明体积的对数是\(X\)列上的次模函数，对于具有足够大奇异值的满列秩矩阵\(X\)，它是非负非减函数。结果表明，传统带列主元的Businger - Golub QR是一种贪心算法，相对误差至多37%；而Gu - Eisenstat强秩揭示QR是一种1 - 交换算法，相对误差至多50%。对于一般矩阵，Businger - Golub QR是最大化\(X\)的QR分解中三角矩阵迹的贪心算法，Gu - Eisenstat QR是1 - 交换算法，前者误差界更好。此结论还扩展到在对称正定矩阵中寻找最大体积的\(k\times k\)子矩阵。
英文摘要
The problem is to select k columns with maximal volume from a real mxn matrix X. We show that the logarithm of the volume is a set submodular function on columns of X, and for full column-rank matrices X with sufficiently large singular values, it is a non-negative non-decreasing function. As a consequence, traditional Businger-Golub QR with column pivoting is a greedy algorithm, with a relative error of at most 37 percent. In contrast, Gu-Eisenstat strong rank-revealing QR is a 1-interchange algorithm, with a relative error of at most 50 percent. The higher accuracy, under this metric, of the simple QR with column pivoting confirms its well known effectiveness in practice. For general, possibly rank-deficient matrices, we derive probabilistic bounds for the absolute error based on a smoothed analysis. The above analyses are extended to finding kxk submatrices of maximal volume in symmetric positive-definite matrices.
