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.

Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing

The Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on the development of artificial intelligence (AI) and other branches of computer science. In this paper, by involving the Wasserstein metric on SPD(n), we obtain computationally feasible expressions for some geometric quantities, including geodesics, exponential maps, the Riemannian connection, Jacobi fields and curvatures, particularly the scalar curvature. Furthermore, we discuss the behavior of geodesics and prove that the manifold is globally geodesic convex. Finally, we design algorithms for point cloud denoising and edge detecting of a polluted image based on the Wasserstein curvature on SPD(n). The experimental results show the efficiency and robustness of our curvature-based methods.

On the Efficiency of Algorithms with Multi-level Parallelism

The paper investigates the efficiency of algorithms for solving computational mathematics problems that use a multilevel model of parallel computing on heterogeneous computer systems. A methodology for estimating the acceleration of algorithms for computers using a multilevel model of parallel computing is proposed. As an example, the parallel algorithm of the iteration method on a subspace for solving the generalized algebraic problem of eigenvalues of symmetric positive definite matrices of sparse structure is considered. For the presented algorithms, estimates of acceleration coefficients and efficiency were obtained on computers of hybrid architecture using graphics accelerators, on multi-core computers with shared memory and multi-node computers of MIMD-architecture.

Numerical Accuracy of Ladder Schemes for Parallel Transport on Manifolds

Parallel transport is a fundamental tool to perform statistics on Riemannian manifolds. Since closed formulae do not exist in general, practitioners often have to resort to numerical schemes. Ladder methods are a popular class of algorithms that rely on iterative constructions of geodesic parallelograms. And yet, the literature lacks a clear analysis of their convergence performance. In this work, we give Taylor approximations of the elementary constructions of Schild’s ladder and the pole ladder with respect to the Riemann curvature of the underlying space. We then prove that these methods can be iterated to converge with quadratic speed, even when geodesics are approximated by numerical schemes. We also contribute a new link between Schild’s ladder and the Fanning scheme which explains why the latter naturally converges only linearly. The extra computational cost of ladder methods is thus easily compensated by a drastic reduction of the number of steps needed to achieve the requested accuracy. Illustrations on the 2-sphere, the space of symmetric positive definite matrices and the special Euclidean group show that the theoretical errors we have established are measured with a high accuracy in practice. The special Euclidean group with an anisotropic left-invariant metric is of particular interest as it is a tractable example of a non-symmetric space in general, which reduces to a Riemannian symmetric space in a particular case. As a secondary contribution, we compute the covariant derivative of the curvature in this space.

WISDoM: Characterizing Neurological Time Series With the Wishart Distribution

WISDoM (Wishart Distributed Matrices) is a framework for the quantification of deviation of symmetric positive-definite matrices associated with experimental samples, such as covariance or correlation matrices, from expected ones governed by the Wishart distribution. WISDoM can be applied to tasks of supervised learning, like classification, in particular when such matrices are generated by data of different dimensionality (e.g., time series with same number of variables but different time sampling). We show the application of the method in two different scenarios. The first is the ranking of features associated with electro encephalogram (EEG) data with a time series design, providing a theoretically sound approach for this type of studies. The second is the classification of autistic subjects of the Autism Brain Imaging Data Exchange study using brain connectivity measurements.