On the distance concentration awareness of certain data reduction techniques

We make a first investigation into a recently raised concern about the suitability of existing data analysis techniques when faced with the counter-intuitive properties of high dimensional data spaces, such as the phenomenon of distance concentration. Under the structural assumption of a generic linear model with a latent variable and an additive unstructured noise, we find that dimension reduction that explicitly guards against distance concentration recovers the well-known techniques of Fisher's linear discriminant analysis, Fisher's discriminant ratio and a variant of projection pursuit. Extrapolation to regression uncovers a close link to sure independence screening, which is a recently proposed technique for variable selection in ultra-high dimensional feature spaces. Hence, these techniques may be seen as distance concentration aware, despite they have not been explicitly designed to have this property. Throughout our analysis, other than the dependency structure implied by the mentioned linear model, we make no assumptions about the distributions of the variables involved.