The interpolative decomposition (ID) aims to construct a low-rank approximation formed by a basis consisting of row/column skeletons in the original matrix and a corresponding interpolation matrix. This work explores fast and accurate ID algorithms from comprehensive perspectives for empirical performance, including accuracy in both skeleton selection and interpolation matrix construction, efficiency in terms of asymptotic complexity and hardware efficiency, as well as rank adaptiveness. While many algorithms have been developed to optimize some of these aspects, practical ID algorithms proficient in all aspects remain absent. To fill in the gap, we introduce robust blockwise random pivoting (RBRP) that is asymptotically fast, hardware-efficient, and rank-adaptive, providing accurate skeletons and interpolation matrices comparable to the best existing ID algorithms in practice. Through extensive numerical experiments on various synthetic and natural datasets, we demonstrate the appealing empirical performance of RBRP from the aforementioned perspectives, as well as the robustness of RBRP to adversarial inputs.

翻译：插值分解（ID）旨在构建一种低秩近似，该近似由原始矩阵的行/列骨架构成的基以及相应的插值矩阵组成。本研究从经验性能的综合视角探索快速且精确的ID算法，包括骨架选取和插值矩阵构建的精确性、渐近复杂度和硬件效率方面的效率，以及秩自适应性。尽管已有许多算法被开发以优化其中某些方面，但能在所有方面都表现出色的实用ID算法仍然缺失。为填补这一空白，我们引入了鲁棒分块随机主元选取（RBRP）算法，该算法具有渐近快速、硬件高效和秩自适应的特点，在实践中能提供与现有最佳ID算法相媲美的精确骨架和插值矩阵。通过对各种合成与自然数据集的大量数值实验，我们从上述角度展示了RBRP引人注目的经验性能，以及RBRP对对抗性输入的鲁棒性。