Tighter bound of Sketched Generalized Matrix Approximation

TL;DR

This paper introduces new sketching techniques that significantly improve the efficiency of generalized matrix approximation algorithms, achieving tighter bounds and better performance for large datasets in machine learning applications.

Contribution

The paper develops novel sketching methods that provide a $(1+ ext{epsilon})$ approximation with smaller sketched dimensions, enhancing the efficiency of matrix approximation.

Findings

Achieves a $(1+ ext{epsilon})$ approximation ratio with smaller sketched dimensions.

Provides a tighter bound on approximation error compared to previous methods.

Improves efficiency for large-scale matrix approximation tasks.

Abstract

Generalized matrix approximation plays a fundamental role in many machine learning problems, such as CUR decomposition, kernel approximation, and matrix low rank approximation. Especially with today's applications involved in larger and larger dataset, more and more efficient generalized matrix approximation algorithems become a crucially important research issue. In this paper, we find new sketching techniques to reduce the size of the original data matrix to develop new matrix approximation algorithms. Our results derive a much tighter bound for the approximation than previous works: we obtain a $(1+\epsilon)$ approximation ratio with small sketched dimensions which implies a more efficient generalized matrix approximation.