发表机构
University of Massachusetts Amherst; University of California Berkeley（马萨诸塞大学阿默斯特分校; 加州大学伯克利分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出一种期望O(n)时间的半正定矩阵快速长度平方采样算法，消除了需获取矩阵列范数的假设，可应用于鲁棒半正定低秩近似等问题，效果接近文献中的复杂方法。
AI 中文摘要
我们提出一种基于简单拒绝采样的算法，用于对n×n半正定（psd）矩阵执行长度平方采样，即按各列的ℓ₂范数平方的概率采样列。该算法的期望运行时间仅为O(n)，远小于输入矩阵规模的亚线性时间，且即便假设输入为对角矩阵，其运行时间仍是最优的。本研究成果有多项应用：多项矩阵问题的亚线性时间算法（如低秩近似、特征值近似）会用到长度平方采样，这类算法通常假设可获取矩阵列范数，因此能高效执行长度平方采样，而本研究表明，至少对于半正定矩阵，可消除该假设。此外，我们还讨论了其在渐近最优算法中的应用，该算法用于估计半正定矩阵的Frobenius范数相对误差。最后，我们证明，本采样算法可得到一种非常简单的亚线性时间算法，用于解决Bakshi等人（FOCS，2020）提出的鲁棒半正定低秩近似问题，其效果几乎与该文献中更复杂的方法相当。
英文摘要
We describe a simple rejection-sampling-based algorithm to perform length-squared sampling on an $n \times n$ positive-semidefinite (psd) matrix: that is, to sample a column with probability proportional to its squared $\ell_2$-norm. The algorithm runs in just $O(n)$ expected time, which is significantly sublinear in the input matrix size. The runtime is optimal, even when the input is assumed to be diagonal. Our result has several applications. Length-squared sampling is used by a number of sublinear time algorithms for matrix problems, like low-rank approximation and eigenvalue approximation. Often, it is assumed that the algorithm is given access to the matrix column norms, and thus can perform length-squared sampling efficiently. Our result shows that, at least for psd matrices, we can remove this assumption. We also discuss an application to an asymptotically optimal algorithm for estimating the Frobenius norm of a psd matrix to relative error. Finally, we show that our sampling algorithm yields a very simple sublinear time algorithm for the robust psd low-rank approximation problem introduced by Bakshi et al. (FOCS, 2020), which nearly matches the more complex method developed there.
