随机选主元Cholesky算法的新分析
A new analysis of the randomly pivoted Cholesky algorithm
AI总结：
本文针对随机选主元Cholesky算法的理论分析缺口，证明其近似误差界与复杂度，还给出高概率误差界，论证主要来自GPT 5.6-Sol (Pro)。
AI中文摘要：
随机选主元Cholesky算法是计算大型半正定矩阵低秩近似的主流方法之一。然而，尽管它在实验中始终能达到与同类竞争方法相当甚至更优的精度，其理论分析却稍落后于其他方法。本文填补了这一空白，证明随机选主元Cholesky算法生成的近似解，其期望误差在最优秩r近似的(1+ε)倍以内，计算步数为O(r/ε + r√log r)。该结果几乎匹配了基于部分Cholesky分解（又称列Nyström近似）的任何低秩近似方法的最优复杂度Θ(r/ε)。本文还给出了高概率成立的随机选主元Cholesky迹误差和谱范数误差的界。该数学论证主要归功于GPT 5.6-Sol (Pro)，作者仅做了部分改进。
英文摘要：
The randomly pivoted Cholesky algorithm is one of the leading methods for computing a low-rank approximation to a large positive-semidefinite matrix. However, while it consistently achieves accuracy comparable to or better than competing methods of its type in experiments, its theoretical analysis lags somewhat behind other methods. This paper closes this gap, proving that randomly pivoted Cholesky produces an approximation with expected error within a $1+\varepsilon$ factor of the optimal rank-$r$ approximation in $\mathcal{O}(r/\varepsilon + r\sqrt{\log r})$ steps. This result nearly matches the optimal complexity $Θ(r/\varepsilon)$ for any low-rank approximation method based on a partial Cholesky decomposition (also known as a column Nyström approximation). The paper also presents bounds on the randomly pivoted Cholesky trace and spectral-norm errors that hold with high probability. The mathematical argument is largely due to GPT 5.6-Sol (Pro), with some refinements by the author.
