New Interpolation Models

David J C MacKay
Cavendish Laboratory
Cambridge
mackay@mrao.cam.ac.uk

Ryo Takeuchi
Waseda University
Tokyo
takeuchi@matsumoto.elec.waseda.ac.jp

Abstract

A traditional linear interpolation model is characterized by three choices: the basis functions used to represent the interpolant; the regularizer applied to the interpolant; and the noise model. Traditionally, the regularizer has a single associated regularization constant $\alpha$, and the noise model has a single parameter $\beta$. In many models the ratio $\alpha/\beta$ alone is responsible for determining globally all the following attributes of the interpolant: its `complexity', `flexibility', `smoothness', `characteristic scale length', `characteristic amplitude', and `power spectrum'. We believe that these terms are not all synonyms. We also suggest that the `smoothness' of an interpolant is not necessarily expected to be the same globally. Rather, we think that interpolation models should be able to capture more than just one flavour of simplicity and complexity. We describe models in which the interpolant may show a smoothness or characteristic amplitude that varies spatially. We emphasize the importance, from the point of view of practical implementation, of the concept of `conditional convexity' when designing models with many hyperparameters.
MaxEnt 94 Abstracts / mas@mrao.cam.ac.uk