  Methods for Projection Pursuit aim to facilitate the visual exploration of
high-dimensional data by identifying interesting low-dimensional projections. A
major challenge is the design of a suitable quality metric of projections,
commonly referred to as the projection index, to be maximized by the Projection
Pursuit algorithm. In this paper, we introduce a new information-theoretic
strategy for tackling this problem, based on quantifying the amount of
information the projection conveys to a user given their prior beliefs about
the data. The resulting projection index is a subjective quantity, explicitly
dependent on the intended user. As a useful illustration, we developed this
idea for two particular kinds of prior beliefs. The first kind leads to PCA
(Principal Component Analysis), shining new light on when PCA is (not)
appropriate. The second kind leads to a novel projection index, the
maximization of which can be regarded as a robust variant of PCA. We show how
this projection index, though non-convex, can be effectively maximized using a
modified power method as well as using a semidefinite programming relaxation.
The usefulness of this new projection index is demonstrated in comparative
empirical experiments against PCA and a popular Projection Pursuit method.
