From Ridge to Lasso - Lasso, Exactly

Lasso keeps the same through-origin rational setup as ridge, but swaps the squared penalty for an absolute-value penalty. The change is exact and visible.

highlighted = computed this step

Same through-origin setup

Use the through-origin model y≈w*x on the exact points (1, 2), (2, 4), and (3, 6). The source sums are sum xy=28 and sum x squared=14.

y\approx wx,\quad \sum xy=28,\quad \sum x^2=14

Swap the penalty

Ridge uses a squared-weight penalty. Lasso swaps in the absolute-value penalty |w|. The objective convention here is one-half times squared error plus λ|w|, so the threshold formula stays clean.

1/2\sum_i(y_i-wx_i)^2+\lambda |w|

Summary

The data and model are exact rationals. Only the penalty changes: Lasso uses |w|, which can make the final weight exactly zero.

\text{same sums; absolute-value penalty}