where the coefficients and ai only depend on the inner products {hxi iwe will see shortly that the representer theorem generalizes this result to broad class of regularized optimization problems we illustrate in figure how the solutions of the ridge regression problems appearing in examples and are qualitatively affected by the regularization parameter for simple linear regression model the data was generated from the model yi - xi ei where each xi is drawn independently and uniformly from the interval [ and each ei is drawn independently from the standard normal distribution = : - - - - - - - - - - - - figure ridge regression solutions for simple linear regression problem each panel shows contours of the loss function (log scaleand the effect of the regularization parameter { }appearing in ( and ( top rowboth terms are penalized bottom rowonly the non-constant term is penalized penalized (plusand unpenalized (diamondsolutions are shown in each case the contours are those of the squared-error loss (actually the logarithm thereof)which is minimized with respect to the model parameters and the diamonds all represent the same minimizer of this loss the plusses show each minimizer [ * * ]of the regularized minimization problems ( and ( for three choices of the regularization parameter for the top three panels the regularization involves both and through the squared norm the circles show the points that have the same squared norm as