A geometric mean algorithm of symmetric positive definite matrices
Juan Olmos, Fabio Martínez, Juan C. Galvis · Revista Colombiana de Matemáticas · 2024
This work introduces a geometric mean algorithm for positive definite matrices using generalized eigenvalue problems and Cholesky factorization. The geometric mean of a finite set of positive definite matrices minimizes the sum of square distances to all the matrices where the distance is an affine-invariant Riemannian metric in the manifold of the symmetric positive definite matrices SN++. In order to compute numerical approximations of the geometric mean several algorithms have been proposed. Some of these algorithms require the computation of several diagonalizations in each iteration. We show that by rewriting the iterations in terms of generalized eigenvalue problems, it is possible to omit some of the diagonalizations at the cost of introducing much less generalized eigenvalue problems that can be solved using Cholesky factorizations. We numerically compare the performance of classical methods and the modified algorithms that use generalized eigenvalue problems. The resulting method is applied to video analysis using the mean of covariance matrices as a compact descriptor for actions classification. The proposed mean descriptor with just 105 scalar values achieved an average accuracy of 75% over a publicaction video dataset.