# 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