in particularfor the squared-error loss we have ar er min (ka qe aka ( this is convex optimization problemand its solution is found by differentiating ( with respect to and and equating to zeroleading to the following system of ( mlinear equations>kkn gk kq ( qkqq as long as is of full column rankthe minimizing function is unique example (ridge regression (cont )we return to example and identify that is the rkhs with linear kernel function (xx xx and is the linear space of constant functions in this caseh is spanned by the function moreoverk xxand if we appeal to the representer theorem directlythen the problem in ( becomesas result of ( ) xxa kxak min , this is convex optimization problemand so the solution follows by taking derivatives and setting them to zero this gives the equations xx(xxn in and ( xxanote that these are equivalent to ( and ( (once again assuming that and has full rank pequivalentlythe solution is found by solving ( )>xx xx xxxx xx xx this is system of ( linear equationsand is typically of much larger dimension than the ( linear equations given by ( and ( as suchone may question the practicality of reformulating the problem in this way howeverthe benefit of this formulation is that the problem can be expressed entirely through the gram matrix kwithout having to explicitly compute the feature vectors -in turn permitting the (implicituse of infinite dimensional feature spaces example (estimating the peaks functionfigure shows the surface plot of the peaks function - -( + ) ( ( - - -( + - ( the goal is to learn the function (xbased on small set of training data (pairs of (xyvaluesthe red dots in the figure represent data {(xi yi )} = where yi (xi and the {xi have been chosen in quasi-random wayusing hammersley points (with bases quasi-random