Title: Affine vs orthogonal equivariance in multivariate analysis
Authors: John Kent - University of Leeds (United Kingdom) [presenting]
Abstract: Many of the classic methods in multivariate analysis (e.g. Hotelling's $T^2$, Fisher's linear discriminant analysis, MANOVA, canonical correlation analysis, independent component analysis) are affine equivariant. That is, they give essentially the same answer if the data undergo an affine transformation. However, other methods, especially those that can be used for high-dimensional data, are only orthogonally equivariant. Examples include PCA, ridge regression, PLS, $k$-means clustering, support vector machines, and projection pursuit. The restriction to orthogonal equivariance is an example of regularization. Some form of regularization is essential if a statistical method is to be applicable in high-dimensional problems. The focus will be on what is gained and what is lost by this restriction. In addition, comparisons will be made to variable-based methods of regularization, such as decision trees and LASSO-type penalties.