L1 Regularization#
- pylit.methods.l1_reg.l1_reg(lambd)#
This is the L1 regularization method. The interface is described in Methods.
The objective function
\[f(u, w, \lambda) = \frac{1}{2} \| \widehat u - \widehat w\|^2_{L^2(\mathbb{R})} + \lambda \| u \|_{L^1(\mathbb{R})},\]is implemented as
\[f(\boldsymbol{\alpha}) = \frac{1}{2} \| \boldsymbol{R} \boldsymbol{\alpha} - \boldsymbol{F} \|^2_2 + \lambda \| \boldsymbol{\alpha} \|_1,\]with the gradient
\[\nabla_{\boldsymbol{\alpha}} f(\boldsymbol{\alpha}) = \boldsymbol{R}^\top(\boldsymbol{R} \boldsymbol{\alpha} - \boldsymbol{F}) \pm \lambda, \quad \boldsymbol{\alpha} \neq 0,\]the learning rate
\[\eta = \frac{1}{\| \boldsymbol{R}^\top \boldsymbol{R} \|}, \quad \boldsymbol{\alpha} \neq 0,\]and the solution
\[\boldsymbol{\alpha}^* = (\boldsymbol{R}^\top \boldsymbol{R})^{-1} (\boldsymbol{R}^\top \boldsymbol{F} \pm \lambda), \quad \boldsymbol{\alpha} \neq 0,\]where
\(\boldsymbol{R}\): Regression matrix,
\(\boldsymbol{F}\): Target vector,
\(\boldsymbol{\alpha}\): Coefficient vector,
\(\lambda\): Regularization parameter,
\(n\): Number of samples.
- Return type: