A similar expression exists if β̃j < 0, and β̃j = 0 is treated separately. Simple calculus shows [Donoho and Johnstone, 1994] that the coordinate-wise update has the form (5)β˜j←S(1N∑i=1Nxij(yi−y˜i(j)),λα)1+λ(1−α) where y˜i(j)=β˜0+∑ℓ≠jxiℓβ˜ℓ is the fitted value excluding the contribution from xij, and hence yi−y˜i(j) the partial residual for fitting βj. Because of the standardization, 1N∑i=1nxij(yi−y˜i(j)) is the simple least-squares coefficient when fitting this partial residual to xij.S(z, γ) is the soft-thresholding operator with value (6)sign(z)(|z|−γ)+={z−γifz>0 and γ<|z|z+γifz<0 and γ<|z|0ifγ≥|z|.
