Direct QR factorizations for tall-and-skinny matrices in MapReduce architectures

TL;DR

This paper introduces a stable and efficient direct QR factorization method for tall-and-skinny matrices in MapReduce environments, improving stability and performance over existing approaches.

Contribution

It presents a new direct TSQR algorithm for MapReduce that is both numerically stable and requires only slightly more than two data passes, unlike previous unstable methods.

Findings

The new TSQR method is more stable than Cholesky QR in MapReduce.

It outperforms the Householder QR method in both theory and practice.

The approach enables efficient SVD computation with minimal data passes.

Abstract

The QR factorization and the SVD are two fundamental matrix decompositions with applications throughout scientific computing and data analysis. For matrices with many more rows than columns, so-called "tall-and-skinny matrices," there is a numerically stable, efficient, communication-avoiding algorithm for computing the QR factorization. It has been used in traditional high performance computing and grid computing environments. For MapReduce environments, existing methods to compute the QR decomposition use a numerically unstable approach that relies on indirectly computing the Q factor. In the best case, these methods require only two passes over the data. In this paper, we describe how to compute a stable tall-and-skinny QR factorization on a MapReduce architecture in only slightly more than 2 passes over the data. We can compute the SVD with only a small change and no difference in performance. We present a performance comparison between our new direct TSQR method, a standard unstable implementation for MapReduce (Cholesky QR), and the classic stable algorithm implemented for MapReduce (Householder QR). We find that our new stable method has a large performance advantage over the Householder QR method. This holds both in a theoretical performance model as well as in an actual implementation.