A new approach to dimensionality reduction: theory and algorithms
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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.