# A Fast Frequent Directions Algorithm for Low Rank Approximation

## TL;DR

This paper introduces a fast, randomized frequent directions algorithm that improves low rank approximation efficiency by integrating sparse subspace embedding, demonstrating strong theoretical guarantees and practical performance.

## Contribution

It presents a novel fast algorithm combining frequent directions with sparse subspace embedding for improved low rank approximation efficiency.

## Key findings

- Produces accurate low rank approximations with linear sketch size
- Demonstrates efficiency on synthetic and real datasets
- Effective in network analysis applications

## Abstract

Recently a deterministic method, frequent directions (FD) is proposed to solve the high dimensional low rank approximation problem. It works well in practice, but experiences high computational cost. In this paper, we establish a fast frequent directions algorithm for the low rank approximation problem, which implants a randomized algorithm, sparse subspace embedding (SpEmb) in FD. This new algorithm makes use of FD's natural block structure and sends more information through SpEmb to each block in FD. We prove that our new algorithm produces a good low rank approximation with a sketch of size linear on the rank approximated. Its effectiveness and efficiency are demonstrated by the experimental results on both synthetic and real world datasets, as well as applications in network analysis.