John Ellipsoids via Lazy Updates

2025-01-03

AI Summary

Computing approximate John ellipsoids for high-dimensional point sets faces a fundamental trade-off among accuracy, computational efficiency, and memory footprint. Method: We propose a novel streaming algorithm that (i) introduces a “lazy update” mechanism for leverage scores, drastically reducing the frequency of high-accuracy recomputation; (ii) integrates importance sampling with batched rectangular matrix multiplication to accelerate iterative refinement; and (iii) employs a low-space-complexity streaming framework enabling scalable processing of large-scale dynamic data. Contribution/Results: Theoretically, our method achieves substantially improved time complexity; practically, it maintains high approximation accuracy even under stringent memory constraints. Empirical evaluation demonstrates speedups of multiple orders of magnitude over state-of-the-art approaches and confirms scalability to ultra-large-scale streaming settings.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Data Stream Mining
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• High-dimensional space
• Ellipse fitting
• Memory management

Innovation

• Approximate Ellipsoid Computation
• Memory-efficient Streaming Technique
• Importance Weighting Algorithm