Data processing on manifolds: Some basic ideas of Riemannian computing with applications

Averaging, distance measurements and finding the shortest connection between two points are among the most basic data processing procedures. However, they form the basis of a multitude of more advanced operations such as data regression, interpolation and optimization. For data sets in a flat Euclidean vector space, the calculations of mean values, shortest paths and interpoint distances are straightforward. However, in applications such as image processing, robot motion planning, dimension reduction, or parametric model reduction, a single datum may be a more complex object, for example, a positive definite covariance matrix, a rotation in 3-space, an orthogonal frame, or an eigenface subspace. For such data, there is no vector space structure. Rather these data objects represent locations on Riemannian manifolds. In this talk, we will outline some of the basic ideas and techniques for data processing on Riemannian manifolds. Special focus will be given on how to obtain explicit formulas for essential geometric quantities on matrix Lie groups and quotients of matrix Lie groups. As numerical examples we will consider Riemannian optimization and interpolation problems.

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