Visualization of Riemannian-manifold-valued elements by multidimensional scaling

Abstract

The present contribution suggests to utilize a multidimensional scaling algorithm as a visualization tool for high-dimensional smoothly constrained learnable-system's patterns that lie on Riemannian manifolds. Such visualization tool proves useful in machine learning whenever learning/adaptation algorithms insist on high-dimensional Riemannian parameter manifolds. In particular, the manuscript describes the cases of interest in the recent scientific literature that the parameter space is the set of special orthogonal matrices, the unit hypersphere and the manifold of symmetric positive-definite matrices. The paper also recalls the notion of multidimensional scaling and discusses its algorithmic implementation. Some numerical experiments performed on toy problems help the readers to get acquainted with the problem at hand, while experiments performed on independent component analysis data as well as averaging data show the usefulness of the proposed visualization tool.