发表机构
University of Florida（佛罗里达大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对随机交替投影方法，提出体积采样的性能界刻画，揭示谱均衡机制，引入高效均匀采样方案，其收敛性优于体积采样且适用于多种迭代方法。
AI 中文摘要
我们研究一类随机交替投影方法，这类方法在由给定集合中n个向量子集张成的子空间之间交替切换。当这些子空间的采样概率与其对应子集内向量张成的体积成正比时，我们推导了性能界的显式刻画。对于每个n，该界通过对关联矩阵的谱进行显式非线性变换得到。我们的分析揭示了一种显式的谱均衡机制，该机制会随着n的增大将谱推向更优的条件数，建立了体积采样与凯莱-哈密顿定理之间的意外联系。此外，我们引入了一种计算高效的均匀采样方案，该方案通过特定的松弛策略实现了可比的理论保证。实验结果表明，这种松弛方法的收敛速度与体积采样相当，且通常优于体积采样，同时避免了体积采样在大规模场景下的不可行性。除了随机Kaczmarz方法，这些结果也直接适用于随机高斯-赛德尔法和坐标下降法。
英文摘要
We study a family of randomized alternating projection methods that alternate among subspaces spanned by subsets of \(n\) vectors from a prescribed set. We derive an explicit characterization of performance bounds when these subspaces are sampled with probabilities proportional to the volumes subtended by the vectors in their corresponding subsets. For each \(n\), the bound is obtained through an explicit nonlinear transformation of the spectrum of an associated matrix. Our analysis reveals an explicit spectral equalization mechanism that drives the spectrum toward improved conditioning as $n$ increases, establishing an unexpected connection between volume sampling and the Cayley-Hamilton theorem. Furthermore, we introduce a computationally efficient uniform sampling scheme that achieves comparable theoretical guarantees through a specific relaxation strategy. Empirical results show that this relaxation method achieves convergence rates comparable to, and often better than, volume sampling, while avoiding the infeasibility of volume sampling at large scale. Besides the randomized Kaczmarz, these results also directly apply to randomized Gauss-Seidel and coordinate descent methods.
