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7/17/2025, 8:57:05 PM
>>16726896
The contours of the objective function are more likely to hit a corner point or ridge of the diamond shape, which will result in both cases in some coefficients being zero, than they are to hit a point of a sphere where there is sparsity. Picrel shows this for the corners, but for the ridges it helps to think in 3d. In 2d only the corner points set coefficients to zero, in 3d the ridges do so as well.
Think of it as the unconstrained minimum location pulling the coefficients to them, while the penalization is pulling the coefficients back to the l1 ball. With the geometric picture in mind, you can see why the constrained solution is more likely to lie on a corner or ridge and these points induce sparsity.
The contours of the objective function are more likely to hit a corner point or ridge of the diamond shape, which will result in both cases in some coefficients being zero, than they are to hit a point of a sphere where there is sparsity. Picrel shows this for the corners, but for the ridges it helps to think in 3d. In 2d only the corner points set coefficients to zero, in 3d the ridges do so as well.
Think of it as the unconstrained minimum location pulling the coefficients to them, while the penalization is pulling the coefficients back to the l1 ball. With the geometric picture in mind, you can see why the constrained solution is more likely to lie on a corner or ridge and these points induce sparsity.
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