Department of Statistics, UCLA

Suppose K₁ , … ,K_m are convex cones in a Hilbert space H, with unit sphere S and inner product ‹ . | . ›. For a particular choice of quantifications, transformations, or representations of a variable x_j in K_j ∩ S we can compute the correlation matrix R(x₁ , … , x_m ) by the rule r_ij (x₁ , … , x_m ) = (x_i | x_j ). Now suppose φ is a real-valued objective function, defined on the space of all correlation matrices. In this paper we study the class of techniques that choose the x_j in their feasible regions K_j ∩ S in such a way that φ(R(x₁ , … , x_m )) is maximized. We discuss typical cases, including linear and nonlinear principal component analysis, canonical correlation analysis, regression analysis. It is shown that correspondence analysis and the Breiman-Friedman ACE-methods are both special cases of this class of techniques. We discuss some choices for the cones K_j , and we indicate that the results simplify greatly if all bivariate regressions can be linearized. A class of iterative projection techniques is suggested, that produces convergent algorithms of simple structure.