The following Representer Theorem and its proof are due to Schölkopf, Herbrich, and Smola: Theorem: Consider a positive-definite real-valued kernel on a non-empty set with a corresponding reproducing kernel Hilbert space. Let there be given
which together define the following regularized empirical risk functional on : Then, any minimizer of the empirical risk admits a representation of the form: where for all. Proof: Define a mapping . Since is a reproducing kernel, then where is the inner product on. Given any, one can use orthogonal projection to decompose any into a sum of two functions, one lying in, and the other lying in the orthogonal complement: where for all. The above orthogonal decomposition and the reproducing property together show that applying to any training point produces which we observe is independent of. Consequently, the value of the error function in is likewise independent of. For the second term, since is orthogonal to and is strictly monotonic, we have Therefore setting does not affect the first term of, while it strictly decreasing the second term. Consequently, any minimizer in must have, i.e., it must be of the form which is the desired result.
Generalizations
The Theorem stated above is a particular example of a family of results that are collectively referred to as "representer theorems"; here we describe several such. The first statement of a representer theorem was due to Kimeldorf and Wahba for the special case in which for. Schölkopf, Herbrich, and Smola generalized this result by relaxing the assumption of the squared-loss cost and allowing the regularizer to be any strictly monotonically increasing function of the Hilbert space norm. It is possible to generalize further by augmenting the regularized empirical risk functional through the addition of unpenalized offset terms. For example, Schölkopf, Herbrich, and Smola also consider the minimization i.e., we consider functions of the form, where and is an unpenalized function lying in the span of a finite set of real-valued functions. Under the assumption that the matrix has rank, they show that the minimizer in admits a representation of the form where and the are all uniquely determined. The conditions under which a representer theorem exists were investigated by Argyriou, Micchelli, and Pontil, who proved the following: Theorem: Let be a nonempty set, a positive-definite real-valued kernel on with corresponding reproducing kernel Hilbert space, and let be a differentiable regularization function. Then given a training sample and an arbitrary error function, a minimizer of the regularized empirical risk admits a representation of the form where for all, if and only ifthere exists a nondecreasing function for which Effectively, this result provides a necessary and sufficient condition on a differentiable regularizer under which the corresponding regularized empirical risk minimization will have a representer theorem. In particular, this shows that a broad class of regularized risk minimizations have representer theorems.
Applications
Representer theorems are useful from a practical standpoint because they dramatically simplify the regularized empirical risk minimization problem. In most interesting applications, the search domain for the minimization will be an infinite-dimensional subspace of, and therefore the search does not admit implementation on finite-memory and finite-precision computers. In contrast, the representation of afforded by a representer theorem reduces the original minimization problem to a search for the optimal -dimensional vector of coefficients ; can then be obtained by applying any standard function minimization algorithm. Consequently, representer theorems provide the theoretical basis for the reduction of the general machine learning problem to algorithms that can actually be implemented on computers in practice. The following provides an example of how to solve for the minimizer whose existence is guaranteed by the representer theorem. This method works for any positive definite kernel, and allows us to transform a complicated optimization problem into a simple linear system that can be solved numerically. Assume that we are using a least squares error function
and a regularization function for some. By the representer theorem, the minimizer
has the form
for some. Noting that
we see that has the form
where and. This can be factored out and simplified to
Since is positive definite, there is indeed a single global minima for this expression. Let and note that is convex. Then, the global minima, can be solved by setting. Recalling that all positive definite matricies are invertible, we see that
