Continuum directions for supervised dimension reduction

We consider dimension reduction of multivariate data under the existence of various types of auxiliary information. We propose a criterion that provides a series of orthogonal directional vectors, that form a basis for dimension reduction. The proposed method can be thought of as an extension from the continuum regression, and the resulting basis is called continuum directions. We show that these directions continuously bridge the principal component, mean difference and linear discriminant directions, thus ranging from unsupervised to fully supervised dimension reduction. With a presence of binary supervision data, the proposed directions can be directly used for a two-group classification. Numerical studies show that the proposed method works well in high-dimensional settings where the variance of the first principal component is much larger than the rest. Primary 60K35, 60K35; secondary 60K35.