Reid Atcheson | Searcharxiv

arXiv2021

A Generalization of QR Factorization To Non-Euclidean Norms

I propose a way to use non-Euclidean norms to formulate a QR-like factorization which can unlock interesting and potentially useful properties of non-Euclidean norms - for example the ability of $l^1$ norm to suppresss outliers or promote sparsity. A classic QR factorization of a matrix $\mathbf{A}$ computes an upper triangular matrix $\mathbf{R}$ and orthogonal matrix $\mathbf{Q}$ such that $\mathbf{A} = \mathbf{QR}$. To generalize this factorization to a non-Euclidean norm $\| \cdot \|$ I relax the orthogonality requirement for $\mathbf{Q}$ and instead require it have condition number $κ\left ( \mathbf{Q} \right ) = \| \mathbf{Q} ^{-1} \| \| \mathbf{Q} \|$ that is bounded independently of $\mathbf{A}$. I present the algorithm for computing $\mathbf{Q}$ and $\mathbf{R}$ and prove that this algorithm results in $\mathbf{Q}$ with the desired properties. I also prove that this algorithm generalizes classic QR factorization in the sense that when the norm is chosen to be Euclidean: $\| \cdot \|=\| \cdot \|_2$ then $\mathbf{Q}$ is orthogonal. Finally I present numerical results confirming mathematical results with $l^1$ and $l^{\infty}$ norms. I supply Python code for experimentation.

math.NA↗

arXiv2019

A Rank Revealing Factorization Using Arbitrary Norms

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.

math.NA↗