A Generalization of QR Factorization To Non-Euclidean Norms

01/24/2021

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by   Reid Atcheson, et al.

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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 𝐀 computes an upper triangular matrix 𝐑 and orthogonal matrix 𝐐 such that 𝐀 = 𝐐𝐑. To generalize this factorization to a non-Euclidean norm · I relax the orthogonality requirement for 𝐐 and instead require it have condition number κ ( 𝐐 ) = 𝐐 ^-1𝐐 that is bounded independently of 𝐀. I present the algorithm for computing 𝐐 and 𝐑 and prove that this algorithm results in 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: ·=·_2 then 𝐐 is orthogonal. Finally I present numerical results confirming mathematical results with l^1 and l^∞ norms. I supply Python code for experimentation.