caa a22 4500 445862025 CHVBK 20180317145446.0 cr unu---uuuuu 170323e20110301xx s 000 0 eng 10.1007/s10092-010-0031-8 doi (NATIONALLICENCE)springer-10.1007/s10092-010-0031-8 Curvature analysis of frequency modulated manifolds in dimensionality reduction [Elektronische Daten] [Mijail Guillemard, Armin Iske] Recent advances in the analysis of high-dimensional signal data have triggered an increasing interest in geometry-based methods for nonlinear dimensionality reduction (NDR). In many applications, high-dimensional datasets typically contain redundant information, and NDR methods are important for an efficient analysis of their properties. During the last few years, concepts from differential geometry were used to create a new range of NDR methods. In the construction of such geometry-based strategies, a natural question is to understand their interaction with classical and modern signal processing tools (convolution transforms, Fourier analysis, wavelet functions). In particular, an important task is the analysis of the incurred geometrical deformation when applying signal transforms to the elements of a dataset. In this paper, we propose the concepts of modulation maps and modulation manifolds for the construction of particular datasets relevant in signal processing and NDR. We consider numerical methods for analyzing geometrical properties of the modulation manifolds, with a particular focus on their scalar and mean curvature. In our numerical examples, we apply the resulting geometry-based analysis to simple test cases, where we present geometrical and topological effects of relevance in manifold learning. Springer-Verlag, 2010 Nonlinear dimensionality reduction nationallicence Manifold learning nationallicence Signal processing nationallicence Fourier and wavelet analysis nationallicence Numerical differential geometry nationallicence Guillemard Mijail Department of Mathematics, University of Hamburg, 20146, Hamburg, Germany aut Iske Armin Department of Mathematics, University of Hamburg, 20146, Hamburg, Germany aut Calcolo Springer Milan 48/1(2011-03-01), 107-125 0008-0624 48:1<107 2011 48 10092 https://doi.org/10.1007/s10092-010-0031-8 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10092-010-0031-8 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Guillemard Mijail Department of Mathematics, University of Hamburg, 20146, Hamburg, Germany aut NATIONALLICENCE 700 1- Iske Armin Department of Mathematics, University of Hamburg, 20146, Hamburg, Germany aut NATIONALLICENCE 773 0- Calcolo Springer Milan 48/1(2011-03-01), 107-125 0008-0624 48:1<107 2011 48 10092 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer