Curvature analysis of frequency modulated manifolds in dimensionality reduction |
| |
Authors: | Mijail Guillemard Armin Iske |
| |
Affiliation: | (1) IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Sachsen, Germany |
| |
Abstract: | 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. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|