Projection Pursuit via Decomposition of Bias Termsof Kernel Density

Dimension reduction of data, < d ! < p (p << d), to be used for clustering has speciic requirements that are not generally met by generic dimension reduction algorithms such as principal components. Projection pursuit, on the other hand, has a growing variety of criteria that target holes, skewness, etc., using information measures, density functionals, sample moments, etc. With the exception of q = 3 by Nason (19xx), the target dimension is only one or two. In this paper, we focus on features of a multivariate kernel density estimator to nd a sequence of basis vectors deening our subspace. We begin by deriving two new variations on the formula for obtaining the asymptotic integrated squared bias (AISB) of a general multivariate kernel density estimator. These formulae are then used to compute orthogonal projections of data onto lower dimensional subspaces in which as much as possible of the AISB is retained. The methods we use for dimension reduction favor those subspaces on which the clustering information of the data is prominent. The methods are not of the 1 \least normal" variety, since the normal density is not the \easiest" estimable density, but rather a certain low order polynomial density.