The Azimuth Project
Blog - Exploring regression on the El Niño data (Rev #2)

Regression using an l 1/2l_{1/2} prior

If we restrict to one variable from the numerous vectors and denote this variable by xx, we get

Ex=Ax+B+sgn(x)2|x|=0 \frac{\partial E}{\partial x} = A x + B + \frac{sgn(x)}{2\sqrt{|x|}} = 0

where AA and BB don’t depend on xx. If we denote sgn(x)/2sgn(x)/2 by CC, we can multiply through by y=|x|y =\sqrt{|x|} to find the minimum (along this co-ordinate) is

±Ay 3+By+C=0 \pm A y^3 + B y + C = 0

where the ±\pm depends whether xx is positive/negative and all subject to needing to ensure the solutions in yy are also consistent with the original equation. Since this is a cubic equation we have a simple closed form for the solutions to this equation and hence can efficiently solve the original equation.