John Ellipsoids via Lazy Updates
This work offers incremental improvements for computational geometry and optimization problems, benefiting researchers in those fields.
The paper tackles the problem of computing approximate John ellipsoids for n points in d dimensions, achieving a faster algorithm by using lazy updates with sampling and fast matrix multiplication, and also provides low-space streaming algorithms.
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.