Improved Practical Matrix Sketching with Guarantees

TL;DR

This paper compares and improves matrix sketching methods, balancing error, size, and computational efficiency, and introduces practical modifications to existing guarantees for better real-world performance.

Contribution

It categorizes known matrix sketching methods, proposes modifications for practical improvements while maintaining guarantees, and provides a reproducible testbed for evaluation.

Findings

iSVD heuristic outperforms known methods in size/error trade-off

Modified FrequentDirections matches iSVD performance with guarantees

Hashing and sampling methods excel in time/error trade-off

Abstract

Matrices have become essential data representations for many large-scale problems in data analytics, and hence matrix sketching is a critical task. Although much research has focused on improving the error/size tradeoff under various sketching paradigms, the many forms of error bounds make these approaches hard to compare in theory and in practice. This paper attempts to categorize and compare most known methods under row-wise streaming updates with provable guarantees, and then to tweak some of these methods to gain practical improvements while retaining guarantees. For instance, we observe that a simple heuristic iSVD, with no guarantees, tends to outperform all known approaches in terms of size/error trade-off. We modify the best performing method with guarantees FrequentDirections under the size/error trade-off to match the performance of iSVD and retain its guarantees. We also demonstrate some adversarial datasets where iSVD performs quite poorly. In comparing techniques in the time/error trade-off, techniques based on hashing or sampling tend to perform better. In this setting we modify the most studied sampling regime to retain error guarantee but obtain dramatic improvements in the time/error trade-off. Finally, we provide easy replication of our studies on APT, a new testbed which makes available not only code and datasets, but also a computing platform with fixed environmental settings.