Title: Big Data, Leverage Scores, and Minimum Volume Covering Ellipsoids: Bridging Theory With Practice
Primary area: optimization
Keywords: Minimum Volume Covering Ellipsoids, D-optimal design, John Ellipsoids, Leverage Scores

Abstract: The Minimum Volume Covering Ellipsoid (MVCE) problem, characterized by $n$ observations in $d$ dimensions where $n \gg d$, can be computationally very expensive in the big data regime. We apply methods from randomized 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 $\mathcal{O}(nd \log n + \text{poly}(d))$, which is a significant improvement in big data problems. Numerical experiments on large-scale synthetic data, as well as real-world data, demonstrate the efficacy of our new algorithm, showing that it substantially reduces computation time while yielding near-optimal solutions.

TL;DR: We use leverage score sampling to reduce computation time for MVCE algorithms