混合精度DEIM-CUR分解的误差分析与精度选择
Error Analysis and Precision Selection for Mixed-Precision DEIM-CUR Decompositions
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中文总结 AI 辅助
本研究放宽了CUR分解中SVD精确计算的假设，分析有限精度算术对DEIM-CUR方法的影响，提出精度选择指导，表明低精度计算不影响近似质量，并通过数值实验验证。
中文摘要 AI 辅助
CUR近似是构造矩阵可解释低秩近似的一种常用方法。构造CUR近似有许多方法，其中许多基于SVD。文献中关于近似质量界限的结果通常依赖于假设SVD的全部或部分被精确计算。在本工作中，我们放宽了这一假设，并展示了近似质量如何受到不精确SVD的影响。此外，我们分析了有限精度算术对DEIM-CUR方法的影响。我们的分析为选择SVD和DEIM索引选择算法的工作精度提供了指导原则。特别是，它表明在许多情况下，可以使用非常低的精度进行这些计算，而不会显著影响近似质量。我们提出了一组数值实验来证明理论结果。
英文摘要
CUR approximation is a popular approach for constructing an interpretable low-rank approximation of a matrix. There are many approaches for constructing the CUR approximation, many of which are based on the SVD. Results in the literature bounding the quality of the approximation usually rely on the assumption that all or part of the SVD is computed exactly. In this work, we relax this assumption and show how the quality of the approximation is affected by an inexact SVD. Moreover, we analyze the effect of finite precision arithmetic on the DEIM-CUR method. Our analysis suggests guidelines for selecting the working precisions for the SVD and the DEIM index selection algorithm. In particular, it suggests that in many cases one can use very low precisions for these computations without significantly impacting the approximation quality. A set of numerical experiments is presented to demonstrate the theoretical results.
发表机构
- Faculty of Mathematics and Physics, Charles University（查理大学数学与物理学院）
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