we see that (xx can be written as the inner product in of the two feature vectors ph(xand ph( )where the feature map ph can be explicitly identified as ph( [ ]thusthe rkhs determined by can be explicitly identified with the space of functions ph( ) for some in the above example we could explicitly identify the feature map howeverin general feature map need not be explicitly available using particular reproducing kernel corresponds to using an implicit (possibly infinite dimensional!feature map that never needs to be explicitly computed representer theorem recall the setting discussed at the beginning of this we are given training data {(xi yi )}ni= and loss function that measures the fit to the dataand we wish to find function that minimizes the training losswith the addition of regularization termas described in section to do thiswe assume first that the class of prediction functions can be decomposed as the direct sum of an rkhs hdefined by kernel function rand another linear space of real-valued functions on xthat isg meaning that any element can be written as with and in minimizing the training loss we wish to penalize the term of but not the term specificallythe aim is to solve the functional optimization problem min loss(yi (xi ) kgk ghh = ( herewe use slight abuse of notationkgkh means khkh if as above in this waywe can view as the null space of the functional kgkh this null space may be emptybut typically has small dimension mfor example it could be the one-dimensional space of constant functionsas in example example (null spaceconsider again the setting of example for which we have feature vectors [ ]and consists of functions of the form xb each function can be decomposed as where xband given gwe have kgkh kbkand so the null space of the functional kgkh (that isthe set of all functions for which kgkh is the set of constant functions herewhich has dimension regularization favors elements in and penalizes large elements in as the regularization parameter varies between zero and infinitysolutions to ( vary from "complex( to "simple( key reason why rkhss are so useful is the following by choosing to be an rkhs in ( this functional optimization problem effectively becomes parametric