John Ellipsoids via Lazy Updates

TL;DR

This paper introduces a faster algorithm for approximating John ellipsoids around points in high-dimensional space by delaying high-accuracy leverage score computations and employing sampling, with applications to streaming algorithms.

Contribution

It significantly speeds up existing algorithms for computing approximate John ellipsoids using lazy updates and sampling techniques.

Findings

Faster algorithm for approximate John ellipsoids.

Efficient low-space streaming algorithms for John ellipsoids.

Reduction in computational complexity through delayed leverage score computation.

Abstract

We give a faster algorithm for computing an approximate John ellipsoid around $n$ points in $d$ dimensions. The best known prior algorithms are based on repeatedly computing the leverage scores of the points and reweighting them by these scores [CCLY19]. We show that this algorithm can be substantially sped up by delaying the computation of high accuracy leverage scores by using sampling, and then later computing multiple batches of high accuracy leverage scores via fast rectangular matrix multiplication. We also give low-space streaming algorithms for John ellipsoids using similar ideas.