L2 Fitting#
- pylit.methods.l2_fit.l2_fit(D, E, lambd)#
This is the L2 fitting method. The interface is described in Methods.
The objective function
\[f(u, w, \lambda) = \frac{1}{2} \| \widehat u - \widehat w\|^2_{L^2(\mathbb{R})} + \frac{1}{2} \lambda \| u - w \|_{L^2(\mathbb{R})}^2,\]is implemented as
\[f(\boldsymbol{\alpha}) = \frac{1}{2} \| \boldsymbol{R} \boldsymbol{\alpha} - \boldsymbol{F} \|^2_2 + \frac{1}{2} \lambda \| \boldsymbol{E} \boldsymbol{\alpha} - \boldsymbol{D} \|^2_2,\]with the gradient
\[\nabla_{\boldsymbol{\alpha}} f(\boldsymbol{\alpha}) = \boldsymbol{R}^\top(\boldsymbol{R} \boldsymbol{\alpha} - \boldsymbol{F}) + \lambda \boldsymbol{E}^\top(\boldsymbol{E} \boldsymbol{\alpha} - \boldsymbol{D}),\]the learning rate
\[\eta = \frac{1}{\| \boldsymbol{R}^\top \boldsymbol{R} + \lambda \boldsymbol{E}^\top \boldsymbol{E} \|},\]and the solution
\[\boldsymbol{\alpha}^* = (\boldsymbol{R}^\top \boldsymbol{R} + \lambda \boldsymbol{E}^\top \boldsymbol{E})^{-1} (\boldsymbol{R}^\top \boldsymbol{F} + \lambda \boldsymbol{E}^\top \boldsymbol{D}),\]where
\(\boldsymbol{R}\): Regression matrix,
\(\boldsymbol{F}\): Target vector,
\(\boldsymbol{E}\): Evaluation matrix,
\(\boldsymbol{D}\): Default model vector,
\(\boldsymbol{\alpha}\): Coefficient vector,
\(\lambda\): Regularization parameter,
\(n\): Number of samples.
- Return type: