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，其基向量由原始矩阵的行/列骨架构成，并配以相应的插值矩阵。本研究从经验性能的综合视角探索快速且精确的ID算法，包括骨架选取与插值矩阵构建的精度、渐进复杂度与硬件效率层面的计算效率，以及秩自适应性。尽管已有诸多算法针对其中某些方面进行优化，但能全面胜任所有方面的实用ID算法仍属空白。为填补这一空白，我们提出了鲁棒分块随机主元选取（RBRP）算法，该算法具有渐进快速性、硬件高效性与秩自适应性，在实践中能提供与现有最优ID算法相媲美的精确骨架与插值矩阵。通过对各类合成与自然数据集进行大量数值实验，我们从上述多个维度展示了RBRP算法令人瞩目的经验性能，并验证了RBRP对对抗性输入具有鲁棒性。