John Ellipsoids via Lazy Updates
David P. Woodruff, Taisuke Yasuda
TL;DR
The paper advances the computation of approximate John ellipsoids for a set of $n$ points in $\mathbb{R}^d$ by combining a fixed-point leverage-score framework with lazy updates and fast rectangular matrix multiplication. It achieves a near-linear runtime in the dense regime, specifically $O\left(\frac{1}{\varepsilon} nd \log(n/d)\right)$ to produce a $Q$ with $\frac{1}{\sqrt{1+\varepsilon}}\,Q\subseteq P\subseteq Q$, and $O\left(\frac{1}{\varepsilon} nd^2 \log(n/d)\right)$ to obtain a $(1+\varepsilon)$-approximation to the MVEE; it also provides low-space streaming algorithms with $O(d^2T)$ space and $T=O(\varepsilon^{-1}\log(n/d))$ passes. A key technical contribution is the use of lazy updates that delay high-accuracy leverage-score computations and rely on fast rectangular matrix multiplication to refresh quadratics in batches, along with accurate probabilistic bounds for products of chi-squared variables and Johnson–Lindenstrauss sketches to control sampling errors. These techniques yield significant speedups over prior leverage-score-reweighting methods and open avenues for nearly linear-time and streaming variants in John ellipsoid 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.