Adaptive randomized pivoting and volume sampling

2025-10-02

AI Summary

This work addresses the Column Subset Selection (CSS) problem by providing a theoretical reformulation and accelerated implementation of the Adaptive Random Pivoting (ARP) algorithm. Methodologically, we establish—for the first time—the rigorous theoretical connections among ARP, volume sampling distributions, and active learning for linear regression, thereby uncovering its implicit sampling mechanism. Leveraging this insight, we design an efficient randomized implementation based on rejection sampling, substantially reducing the sampling time complexity. We also derive tighter approximation error bounds and stronger statistical guarantees, enhancing the algorithm’s theoretical foundation. Empirically, the proposed method achieves speedups of several-fold in sampling time while preserving approximation accuracy comparable to the original ARP. This work not only improves the practicality of ARP but also introduces a novel conceptual framework for understanding the intrinsic links between randomized CSS algorithms and active learning.

Technology Category

• Machine Learning: Active Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Adaptive randomized pivoting (ARP) is a recently proposed and highly effective algorithm for column subset selection. This paper reinterprets the ARP algorithm by drawing connections to the volume sampling distribution and active learning algorithms for linear regression. As consequences, this paper presents new analysis for the ARP algorithm and faster implementations using rejection sampling.

Problem

• Reinterprets adaptive randomized pivoting via volume sampling
• Presents new analysis for column subset selection algorithm
• Develops faster implementations using rejection sampling techniques

Innovation

• Adaptive randomized pivoting for column selection
• Connects volume sampling to active learning
• Faster implementation using rejection sampling