通过条件行列式点过程进行增量列子集选择
Incremental Column Subset Selection via Conditional Determinantal Point Processes
AI总结：
本文提出条件DPPs框架及多阶段ARP算法，实现增量列子集选择，在保证理论误差界的同时达到与标准ARP相当的精度。
AI中文摘要：
列子集选择旨在寻找一组能够准确近似给定矩阵的小规模代表性列。我们从行列式点过程（DPPs）及其与若干现有算法关系的角度研究该问题。为支持增量列选择，我们引入了条件DPPs，该方法可应用于任何列选择策略。特别地，我们将其与自适应随机主元法（ARP）相结合，开发了多阶段ARP（MSARP）算法。我们为这些方法建立了关于期望Frobenius范数近似误差的理论保证。此外，我们提出了ARP的一种固定精度变体，可自适应地确定所选列的数量。关于列子集选择和Nyström近似的数值实验表明，MSARP在实现增量选择的同时，达到了与标准ARP相当的精度。
英文摘要：
Column subset selection aims to seek a small set of representative columns that accurately approximates a given matrix. We study this problem from the perspective of determinantal point processes (DPPs) and their relation to several existing algorithms. To support incremental column selection, we introduce conditional DPPs, which can be applied to any column selection strategy. In particular, we combine it with adaptive randomized pivoting(ARP), and develop a multi-stage ARP (MSARP) algorithm. We establish theoretical guarantees on the expected Frobenius-norm approximation errors for these methods. In addition, we propose a fixed-precision variant of ARP that adaptively determines the number of selected columns. Numerical experiments on column subset selection and Nyström approximation show that MSARP achieves accuracy comparable to standard ARP while enabling incremental selection.
