Geometrical interpretation of L1 regression

by Álvaro Méndez Civieta   Last Updated May 20, 2018 18:19 PM

I have found the following image (or a similar version) in a lot of books related to penalized linear models. I get the insight of this image. The ellipsoids are the solution of the linear regression model as the $\beta$ moves away from the minimum, and the blue areas are the constriants of the penalization, so the solution will be the first point in which the red ellipsoids touch the blue area. My question is: Is there any similar geometrical interpretation for the L1 regression (or the general version, quantile regression)? My intuition sugggests that L1 regression may be a polyhedral, but I am not sure and I have not found anything about this in the books

enter image description here

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