Finite Element Thin Plate Splines for Data Mining Applications

Thin plate splines have been used successfully to model curves and surfaces. A new application is in data mining where they are used to model interaction terms. These interaction splines break the \curse of dimensionality" by reducing the high-dimensional nonparametric regression problem to the determination of a set of interdependent surfaces. However, the determination of the corresponding thin plate splines requires the solution of a dense linear system of equations of order n where n is the number of observations. For data mining applications n can be in the millions, and so standard thin plate splines, even using fast algorithms may not be practical. A nite element approximation of the thin plate splines will be described. The method uses H elements in a formulation which only needs rst order derivatives. The resolution of the method is chosen independently of the number of observations which only need to be read from secondary storage once and do not need to be stored in memory. The formulation leads to a saddle point problem. Convergence and solution of the method and its relationship to the standard thin plate splines will be discussed.