# Updating QR decomposition for geometrically similar least squares problem

Let's say we have a weighted least squares problem under the same matrix $$A$$ such that,

$$\hat{x} := \arg \min_x ||A x - b||_{W_1}$$ where $$|| \cdot ||_{W_1}$$ is the Euclidean norm weighted by diagonal elements in $$W_1$$. We can solve this easily by performing a $$QR$$ decomposition of $$W^{1/2}_1 A$$ together with $$W^{1/2}_1 b$$ using standard software.

Given this, is there an approach to re-use the previous QR decomposition to solve a similar weighted least square problems under the same $$A$$, such that $$\hat{z} := \arg \min_z ||A z - c||_{W_2}$$ if we can bound $$||c - b||$$ and $$||W_1 - W_2||_F$$?

The motivation for this, is to see if there are faster means of iterative re-weighted least squares to solve generalized inverse problems, rather than re-computing a complete $$QR$$ decomposition at each step.

My gut tells me that there is no easy way to re-use a $$QR$$, no matter the bound, as this would make too many related problems easy to solve (and thus folks would already be talking about it). Nature is uncaring in that way.

• How large is your $A$ matrix? Is it sparse? Oct 3, 2023 at 4:22
• Not necessarily large. $n \times p$ for $n \approx 1000$ and $p \approx 25$ (guestimates). Not sparse in general. Oct 4, 2023 at 16:32

Perhaps only some values change, in which case you might be able to make use of the Woodbury matrix identity to make a cheap rank-k update, given that $$\hat{z} = \arg \min_z ||A z - c||_{W}$$ has the solution $$\hat{z} = (A^TWA)^{-1}A^TWc$$