AI 中文总结
该研究针对选主元QR与LU分解，证明近似贪心选主元下误差由子矩阵行列式控制，建立了代数、几何奇异值衰减下的收敛速率，并将LU分析扩展到二元函数。
AI 中文摘要
选主元QR分解与选主元LU分解是用于从选定列或选定行和列计算矩阵低秩近似的贪心算法。尽管它们在实际应用中表现稳健，但将其误差与对应最佳低秩近似误差比较的一般最坏情况界包含指数增长因子，无法解释其在适度奇异值衰减下的表现。我们证明，在近似贪心选主元下，其误差由子矩阵的行列式控制，该行列式被主导奇异值的几何均值所界定。利用此界，我们建立了在代数和几何奇异值衰减下的收敛速率。我们还将LU分析扩展到二元函数，通过界定任意采样子矩阵的行列式，在可微性假设下获得代数收敛速率，在解析性下获得几何收敛速率。
英文摘要
Pivoted QR and pivoted LU decompositions are greedy algorithms used to compute low-rank approximations of matrices from selected columns, or selected rows and columns. Despite their practical robustness, general worst-case bounds comparing their errors with those of the best corresponding low-rank approximations contain exponentially growing factors and do not explain their behavior under modest singular value decay. We prove that under approximate greedy pivoting, their error is controlled by the determinant of a submatrix, which is bounded by the geometric mean of the leading singular values. Using this bound, we establish convergence rates under algebraic and geometric singular value decay. We also extend the LU analysis to functions of two variables. By bounding the determinants of arbitrary sampled submatrices, we obtain algebraic convergence rates under differentiability assumptions and geometric convergence under analyticity.
