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Download the Paper (258 K, PDF file) - October 30, 2005 Tell a colleague about it. Printing Tips: Select 'print as image' in the Acrobat print dialog if you have trouble printing.

ABSTRACT:

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.