A General Least Squares Regression Framework on Matrix Manifolds for Computer Vision

Abstract

Least squares regression is one of the fundamental approaches for statistical analysis. And yet, its simplicity has often led to researchers overlooking it for complex recognition problems. In this chapter, we present a nonlinear regression framework on matrix manifolds. The proposed method is developed based upon two key attributes: underlying geometry and least squares fitting. The former attribute is vital since geometry characterizes the space for classification while the latter exhibits a simple estimation model. Considering geometric properties, we formulate the least squares regression as a composite function. The proposed framework can be naturally generalized to many matrix manifolds. We show that this novel formulation is applicable to matrix Lie groups, SPD(n), Grassmann manifolds, and the product of Grassmann manifolds for a number of computer vision applications ranging from visual tracking, object categorization, and activity recognition to human interaction recognition. Our experiments reveal that the proposed method yields competitive performance, including state-of-the-art results on challenging activity recognition benchmarks.