Armin Iske, University of Hamburg, Germany
This application-oriented tutorial discusses basic tools and techniques concerning the analysis of high-dimensional signal data. To this end, we first explain relevant mathematical methods from signal data analysis, before a more comprehensive discussion on linear and nonlinear projection methods for dimensionality reduction is provided, where particular emphasis is placed on recent concepts from manifold learning. The purpose of our construction is to obtain suitable low-dimensional parameterizations of high-dimensional signal data.
This requires analyzing the geometrical distortion of manifolds, as incurred by their corresponding(nonlinear) embedding maps. To this end, a more detailed discussion concerning the curvature analysis of manifolds is provided, where the computation of their related metric and curvature tensors is explained.
Finally, numerical examples concerning low-dimensional parameterizations of scale- and frequency-modulated manifolds are presented for illustration.