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## Main Question or Discussion Point

I am trying to understand the difference between L1 vs. L2 regularization in OLS. I understand the concept of center of ellipsoid being the optimal solution and ellipse itself being contours of constant squared errors. And when we use L2 regularization we introduce a spherical constraint on coefficient and when we use L1 the constraints are rectangle in R

In all corresponding pictorial representation of the above in literature,etc, the representation in R

Thanks

^{2}representation.In all corresponding pictorial representation of the above in literature,etc, the representation in R

^{2}always shows ellipsoid intersecting the circle in first quadrant but the square on one of the axis i.e at corners. How come in L1 regularization the ellipsoid intersects the square only on corners but in case of L2 any point on the sphere. Wouldn't we get a sparse solution for L2 as well if ellipse intersects the circle (R^{2}representation) at axis.Thanks