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For least squares regression there exists the Recursive Least Squares algorithm which allows to find the least squares solution online.

Does something similar exist for Least Absolute Deviation regression (such that data points can be added and the previous solution can be used such that the solution for the new extended problem does not need to be computed from scratch)?

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    $\begingroup$ I doubt it because there isn't a nice closed form solution like there is in OLS. This looks like a nice survey that might say what I'm guessing explicitly. I didn't read it. mtholyoke.edu/courses/kpepe/Stats%20webpage%20articles/… $\endgroup$
    – mlofton
    Commented Jun 14, 2020 at 19:52
  • $\begingroup$ @mlofton Thank you, that's a good survey! $\endgroup$
    – phinz
    Commented Jun 15, 2020 at 6:11
  • $\begingroup$ Your welcome. As far as I can tell, it doesn't mention the possibility in there so my best guess is that the answer is no. $\endgroup$
    – mlofton
    Commented Jun 16, 2020 at 14:23

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The problem is different from OLS since LAD generates regression hyperplanes that necessarily fit m data points (in m-dimensional space). So, upon the arrival of a new point, LAD will either “jump” to a new hyperplane or stay where it was. This gives it a distinct discreet flavor. I imagine this in a 3D animation where points arrive and I watch the regression plane jump all around. (Someone should code this animation!) The best I’ve seen so far is Narula & Wellington 1985 “Interior Analysis for the Minimum Sum of Absolute Errors Regression” where they talk about the opposite: removing points in a LAD regression … but you’ll get the idea.

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