# A Rank Revealing Factorization Using Arbitrary Norms

## TL;DR

This paper generalizes the rank-revealing QR factorization to arbitrary norms, enabling low-rank approximations with different error metrics, including the $l^1$ norm, and provides practical Python implementation.

## Contribution

It introduces a generalized QR factorization framework for arbitrary norms and demonstrates its application to $l^1$ norm low-rank approximation.

## Key findings

- Generalized QR factorization for any norm with analogous properties.
- Application to $l^1$ norm low-rank approximation.
- Provided Python code for practical implementation.

## Abstract

The classic rank-revealing QR factorization factorizes a matrix $A$ as $AP=QR$ where $P$ permutes the columns of $A$, $Q$ is an orthogonal matrix, and $R$ is upper triangular with non-increasing diagonal entries. This is called rank-revealing because careful choice of $P$ allows the user to truncate the factorization for a low-rank approximation of $A$ with an error term computed in the $l^2$ norm. In this paper I generalize the QR factorization to use any arbitrary norm and prove analogous properties for $Q$ and $R$ in this setting. I then show an application of this algorithm to compute low-rank approximations to $A$ with error term in the $l^1$ norm instead of the $l^2$ norm. I provide Python code for the $l^1$ case as demonstration of the idea.