The aim of the present research work is to investigate algorithms to compute empirical averages of finite sets of sample-points over the Stiefel manifold by extending the notion of Pythagoras' arithmetic averaging over the real line to a curved manifold. The idea underlying the developed algorithms is that sample-points on the Stiefel manifold get mapped onto a tangent space, where the average is taken, and then the average point on the tangent space is brought back to the Stiefel manifold, via appropriate maps. Numerical experimental results are shown and commented on in order to illustrate the numerical behaviour of the proposed procedure. The obtained numerical results confirm that the developed algorithms converge steadily and in a few iterations and that they are able to cope with relatively large-size problems.

Recently we introduced the concept of neural network learning on Stiefel-Grassman manifold for multilayer perceptron—like networks. Contributions of other authors have also appeared in the scientific literature about this topic. This article presents a general theory for it and illustrates how existing theories may be explained within the general framework proposed here.

Averaging is a common way to alleviate errors and random fluctuations in measurements and to smooth out data. Averaging also provides a way to merge structured data in a smooth manner. The present paper describes an algorithm to compute averages on matrix Lie groups. In particular, we discuss the case of averaging over the special orthogonal group of matrices, the unitary group of matrices and the group of symmetric positive-definite matrices.

The prospect and potentiality of interfacing minds with machines has long captured human imagination. Recent advances in biomedical engineering, computer science, and neuroscience are making brain–computer interfaces a reality, paving the way to restoring and potentially augmenting human physical and mental capabilities. Applications of brain–computer interfaces are being explored in applications as diverse as security, lie detection, alertness monitoring, gaming, education, art, and human cognition augmentation. The present tutorial aims to survey the principal features and challenges of brain–computer interfaces (such as reliable acquisition of brain signals, filtering and processing of the acquired brainwaves, ethical and legal issues related to brain–computer interface (BCI), data privacy, and performance assessment) with special emphasis to biomedical engineering and automation engineering applications. The content of this paper is aimed at students, researchers, and practitioners to glimpse the multifaceted world of brain–computer interfacing.