A Sherman--Morrison--Woodbury approach to solving least squares problems with low-rank updates

TL;DR

This paper introduces a fast method to update the pseudoinverse of a matrix after low-rank changes, significantly speeding up solutions to modified least squares problems.

Contribution

It presents a simple formula for updating the pseudoinverse under low-rank modifications, enabling faster least squares solutions.

Findings

The proposed method is computationally efficient.

It outperforms traditional approaches in speed.

Applicable to various low-rank update scenarios.

Abstract

We present a simple formula to update the pseudoinverse of a full-rank rectangular matrix that undergoes a low-rank modification, and demonstrate its utility for solving least squares problems. The resulting algorithm can be dramatically faster than solving the modified least squares problem from scratch, just like the speedup enabled by Sherman--Morrison--Woodbury for solving linear systems with low-rank modifications.