Multivariate Multilinear Regression

Conventional regression methods, such as multivariate linear regression (MLR) and its extension principal component regression (PCR), deal well with the situations that the data are of the form of low-dimensional vector. When the dimension grows higher, it leads to the under sample problem (USP): the dimensionality of the feature space is much higher than the number of training samples. However, little attention has been paid to such a problem. This paper first adopts an in-depth investigation to the USP in PCR, which answers three questions: 1) Why is USP produced? 2) What is the condition for USP, and 3) How is the influence of USP on regression. With the help of the above analysis, the principal components selection problem of PCR is presented. Subsequently, to address the problem of PCR, a multivariate multilinear regression (MMR) model is proposed which gives a substitutive solution to MLR, under the condition of multilinear objects. The basic idea of MMR is to transfer the multilinear structure of objects into the regression coefficients as a constraint. As a result, the regression problem is reduced to find two low-dimensional coefficients so that the principal components selection problem is avoided. Moreover, the sample size needed for solving MMR is greatly reduced so that USP is alleviated. As there is no closed-form solution for MMR, an alternative projection procedure is designed to obtain the regression matrices. For the sake of completeness, the analysis of computational cost and the proof of convergence are studied subsequently. Furthermore, MMR is applied to model the fitting procedure in the active appearance model (AAM). Experiments are conducted on both the carefully designed synthesizing data set and AAM fitting databases verified the theoretical analysis.