Remarks on Functional Canonical Variates, Alternating Least Squares Methods and Ace

We discuss properties of some data-analytic methods which are intimately related to each other: alternating least squares (ALS), correspondence analysis and more recently Breiman and Friedman's ACE algorithm. The application of these methods to regression produces nonparametric estimators of nonlinear transformations, both of the response and the predictors. These procedures are among the most powerful tools for data analysis, but missing awareness of some artifacts could lead to inappropriate interpretations. We point out some anomalies as well as some curiosities in the mathematics of these methods, and we relate them to some areas in computer-aided tomography, projection pursuit regression and nonlinear devices in the theory of noise.