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 $\kappa \left ( \mathbf{Q} \right ) = \| \mathbf{Q} ^{-1} \| \| \mathbf{Q} \|$ 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.
翻译:我提出一种方法来使用非欧洲的规范来制定 QR 类的参数化, 它可以解开非欧洲的规范中有趣和潜在有用的属性。 例如, $1$ 的规范能够抑制外部值或促进线性。 一个矩阵$\ mathbf{A} 的典型 QR 因子化 $\ mathbf{R} 计算一个上三角矩阵$\ mathbf{R} 和 orthonal 矩阵 $\ mathb} 美元( mathb} maclb} licbb} = mathbralb} $_r_r_rxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx