Interpolation and polynomial fitting in the SPD manifold

Abstract

Generalizing to Riemannian manifolds classical methods to approximate data (e.g. averaging, interpolation and regularization) has been a theoretical challenge that has also revealed to be computationally very demanding and often unsatisfactory. One particular manifold that shows up in numerous scientific areas that use tensor analysis, including machine learning, medical imaging, and optimization, is the set of symmetric positive definite (SPD) matrices. In this work, we show that when the SPD matrices are endowed with the Log-Euclidean framework, certain optimization problems, such as interpolation and best fitting polynomial problems, can be solved explicitly. This contrasts with what happens in general non-Euclidean spaces. In the Log-Euclidean framework, the SPD manifold has the structure of a commutative Lie group and when equipped with the Log-Euclidean metric it becomes a flat Riemannian manifold. Explicit expressions for polynomial curves in the SPD manifold are therefore obtained easily, and this enables the complete resolution of the proposed problems.