A Non-Iterative Method for the Difference of Means on the Lie Group of Symmetric Positive-Definite Matrices

A non-iterative method for the difference of means is presented to calculate the log-Euclidean distance between a symmetric positive-definite matrix and the mean matrix on the Lie group of symmetric positive-definite matrices. Although affine-invariant Riemannian metrics have a perfect theoretical framework and avoid the drawbacks of the Euclidean inner product, their complex formulas also lead to sophisticated and time-consuming algorithms. To make up for this limitation, log-Euclidean metrics with simpler formulas and faster calculations are employed in this manuscript. Our new approach is to transform a symmetric positive-definite matrix into a symmetric matrix via logarithmic maps, and then to transform the results back to the Lie group through exponential maps. Moreover, the present method does not need to compute the mean matrix and retains the usual Euclidean operations in the domain of matrix logarithms. In addition, for some randomly generated positive-definite matrices, the method is compared using experiments with that induced by the classical affine-invariant Riemannian metric. Finally, our proposed method is applied to denoise the point clouds with high density noise via the K-means clustering algorithm.