Machine Learning

Appendix B

Regularization Operators

Kernel machines can be nicely presented within a regularization framework based on differential operators. Here we give an introduction to differential and pseudo-differential operators. A natural way of imposing the development of a “smooth solution” f of a learning problem is to think of a special expression of the parsimony principle which relies on restricting the quick variations of f.

In the simplest case in which $f:\mathcal{X}\subset \mathbb{R}\to \mathbb{R}$ and $f\in L^2(\mathcal{X})$, one can introduce the index

$\begin{array}{cc}\hfill \mathcal{R}& =\underset{\mathcal{X}}{\int }{[f^{\prime }(x}^{}\hfill \end{array}$