Lasso

The Lasso minimizes regression error plus an ${\ell }^1$ penalty.

The Dantzig selector minimizes the coefficient ${\ell }^1$ norm subject to an ${\ell }^{\infty }$ bound on residual correlations.

The square-root Lasso uses the unsquared residual norm plus an ${\ell }^1$ penalty.

Soft thresholding maps $u$ to $\mathrm{sgn}(u)(|u|-\lambda {)}_{+}$ and is the proximal operator of the absolute-value penalty.