🤖 AI Summary
Computing exact leverage scores for Kronecker-product structured matrices in large-scale least-squares problems is computationally prohibitive, while existing approximation methods incur statistical bias and high overhead.
Method: We propose the first efficient exact leverage score algorithm tailored to Kronecker-structured matrices. Leveraging the inherent tensor structure, our method designs a near-linear-time framework for exact leverage score computation and sampling—bypassing costly full-matrix SVD or biased sketching approximations.
Contribution/Results: Theoretically and empirically, our algorithm achieves significantly lower sampling error than state-of-the-art approximate methods (e.g., FJLT- or CountSketch-accelerated approaches), while maintaining substantially lower time complexity than full SVD. This work establishes the first scalable, exact, and efficient leverage score sampling scheme for Kronecker-structured matrices, enabling improved structured random projections and large-scale regression.
📝 Abstract
While leverage score sampling provides powerful tools for approximating solutions to large least squares problems, the cost of computing exact scores and sampling often prohibits practical application. This paper addresses this challenge by developing a new and efficient algorithm for exact leverage score sampling applicable to matrices that are lower column subsets of Kronecker product matrices. We synthesize relevant approximation guarantees and detail the algorithm that specifically leverages this structural property for computational efficiency. Through numerical examples, we demonstrate that utilizing efficiently computed exact leverage scores via our methods significantly reduces approximation errors, as compared to established approximate leverage score sampling strategies when applied to this important class of structured matrices.
