Efficient Data-Driven Leverage Score Sampling Algorithm for the Minimum Volume Covering Ellipsoid Problem in Big Data

TL;DR

This paper introduces a novel leverage score sampling algorithm for the MVCE problem in big data, significantly reducing computational complexity while maintaining solution quality, validated through theoretical bounds and numerical experiments.

Contribution

Develops a data-driven leverage score sampling algorithm for MVCE with theoretical error bounds and improved computational complexity in big data settings.

Findings

Reduces MVCE computation from O(nd^2) to O(nd + poly(d))

Achieves near-optimal solutions with less computation time

Theoretically guarantees convergence and error bounds

Abstract

The Minimum Volume Covering Ellipsoid (MVCE) problem, characterised by $n$ observations in $d$ dimensions where $n \gg d$, can be computationally very expensive in the big data regime. We apply methods from randomised numerical linear algebra to develop a data-driven leverage score sampling algorithm for solving MVCE, and establish theoretical error bounds and a convergence guarantee. Assuming the leverage scores follow a power law decay, we show that the computational complexity of computing the approximation for MVCE is reduced from $\mathcal{O}(nd^2)$ to $\mathcal{O}(nd + \text{poly}(d))$, which is a significant improvement in big data problems. Numerical experiments demonstrate the efficacy of our new algorithm, showing that it substantially reduces computation time and yields near-optimal solutions.