## Curvature Analysis of Frequency Modulated Manifolds in Dimensionality Reduction

Citations: | 4 - 4 self |

### BibTeX

@MISC{Guillemard_curvatureanalysis,

author = {Mijail Guillemard and Armin Iske},

title = {Curvature Analysis of Frequency Modulated Manifolds in Dimensionality Reduction},

year = {}

}

### OpenURL

### 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 whole 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 frequency modulation maps and modulation manifolds for the construction of particular datasets relevant in signal processing and NDR. Moreover, we design a numerical algorithm for analyzing geometrical properties of the modulation manifolds, with a particular focus on their scalar curvature. Finally, in our numerical examples, we apply the resulting geometry-based analysis algorithm to two model problems, where we present geometrical and topological effects of relevance in manifold learning.

### Citations

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Citation Context ...etry are an emergent and very active research topics. Basic concepts in this field are the angle defect [2,7] and the usage of the Laplacian 3operator when computing the mean and Gaussian curvatures =-=[1, 3, 11, 19]-=-. A series of additional important developments for manifold sampling have also been developed over the last years [22,23]. Other important topics are generalizations of the curvature concept in Alexa... |

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Citation Context ...erential geometry play an important role [2,7,8,13]. The geometry-based approach of NDR can be viewed as a complementary strategy to statistical oriented methods from machine learning and data mining =-=[5]-=-. ∗ guillemard@math.uni-hamburg.de † iske@math.uni-hamburg.de 1To briefly describe the basic problem of NDR and manifold learning, suppose we are given a dataset X = {xi} m i=1 ⊂ R n lying in a high-... |

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37 | Discrete Laplace Operator for Meshed Surfaces
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Citation Context ...etry are an emergent and very active research topics. Basic concepts in this field are the angle defect [2,7] and the usage of the Laplacian 3operator when computing the mean and Gaussian curvatures =-=[1, 3, 11, 19]-=-. A series of additional important developments for manifold sampling have also been developed over the last years [22,23]. Other important topics are generalizations of the curvature concept in Alexa... |

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28 |
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Citation Context ...for manifold sampling have also been developed over the last years [22,23]. Other important topics are generalizations of the curvature concept in Alexandrov spaces or cell-complexes, as discussed in =-=[9,21]-=-. 2.2 Application Examples Relevant motivations of our framework are time-frequency representations, where a segmentation of a signal is analyzed with Fourier or wavelet functions. For instance, a typ... |

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Citation Context ...M. Due to the Whitney embedding theorem (which states that any connected smooth p-dimensional manifold can smoothly be embedded in R 2p+1 ), one basic condition in this problem is 2p + 1 ≤ d ≤ n, see =-=[1]-=-. Throughout this paper, we use the term manifold to denote a compact smooth connected manifold embedded in the Euclidean space R n . Now a crucial requirement in manifold learning is to ensure condit... |

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Citation Context ...etry are an emergent and very active research topics. Basic concepts in this field are the angle defect [2,7] and the usage of the Laplacian 3operator when computing the mean and Gaussian curvatures =-=[1, 3, 11, 19]-=-. A series of additional important developments for manifold sampling have also been developed over the last years [22,23]. Other important topics are generalizations of the curvature concept in Alexa... |

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Citation Context ...exity of new challenging application problems. In the design of these modern tools, special emphasis is placed on geometrical aspects, where concepts from differential geometry play an important role =-=[2,7,8,13]-=-. The geometry-based approach of NDR can be viewed as a complementary strategy to statistical oriented methods from machine learning and data mining [5]. ∗ guillemard@math.uni-hamburg.de † iske@math.u... |

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Citation Context ...exity of new challenging application problems. In the design of these modern tools, special emphasis is placed on geometrical aspects, where concepts from differential geometry play an important role =-=[2,7,8,13]-=-. The geometry-based approach of NDR can be viewed as a complementary strategy to statistical oriented methods from machine learning and data mining [5]. ∗ guillemard@math.uni-hamburg.de † iske@math.u... |

10 |
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Citation Context ...f weaker structural assumptions on Ω and M is a very valuable but challenging task for a large variety of applications. In this context, the work on persistent homology [14] and discrete Morse theory =-=[3]-=- offers a suitable background that can be used in future steps. 6 Acknowledgments The authors are supported by the priority program DFG-SPP 1324 of the Deutsche Forschungsgemeinschaft (DFG). 12Refere... |

9 | Sampling and Reconstruction of Surfaces and Higher Dimensional Manifolds
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Citation Context ...mation retracts to M, and therefore the homology of U equals the homology of M. A series of additional important developments concerning conditions for efficient sampling of manifolds can be found in =-=[10,11]-=-. 2.1 Application Examples Relevant applications for our investigations are in time-frequency analysis, where a signal is segmented in time consecutive sections, which are then Fourier transformed. Th... |

8 | Bochner’s method for cell complexes and combinatorial ricci curvature
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Citation Context ...for manifold sampling have also been developed over the last years [22,23]. Other important topics are generalizations of the curvature concept in Alexandrov spaces or cell-complexes, as discussed in =-=[9,21]-=-. 2.2 Application Examples Relevant motivations of our framework are time-frequency representations, where a segmentation of a signal is analyzed with Fourier or wavelet functions. For instance, a typ... |

6 | H.: Fast manifold learning based on riemannian normal coordinates
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Citation Context ...ta and signal analysis. In the design of these modern tools, special emphasis is placed on geometrical aspects, where concepts from differential geometry and algebraic topology play an important role =-=[5,6,17,18,25,26]-=-. The geometry-based approach of NDR can be viewed as a complementary strategy to statistical oriented methods from machine learning and data mining [14]. ∗ guillemard@math.uni-hamburg.de Revised vers... |

5 |
Metric Structures in Differential Geometry
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Citation Context ...alar curvature distortion. In this section, we explain the basic ingredients for computing the scalar curvature S of a Riemannian manifold M. As a starting point we regard a metric tensor field for M =-=[6,12]-=-, being defined for a particular system of local coordinates (θi,...,θk), as gij(x) = gij(θi,...,θk) =< ∂i,∂j > . An invariant of a Riemannian manifold with respect to isometries are its sectional cur... |

4 | Geometric Sampling of Manifolds for Image Representation and
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(Show Context)
Citation Context ...mation retracts to M, and therefore the homology of U equals the homology of M. A series of additional important developments concerning conditions for efficient sampling of manifolds can be found in =-=[10,11]-=-. 2.1 Application Examples Relevant applications for our investigations are in time-frequency analysis, where a signal is segmented in time consecutive sections, which are then Fourier transformed. Th... |

3 |
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Citation Context ...exity of new challenging application problems. In the design of these modern tools, special emphasis is placed on geometrical aspects, where concepts from differential geometry play an important role =-=[2,7,8,13]-=-. The geometry-based approach of NDR can be viewed as a complementary strategy to statistical oriented methods from machine learning and data mining [5]. ∗ guillemard@math.uni-hamburg.de † iske@math.u... |

1 | A characterization of the angle defect and the Euler characteristic in dimension 2 - Bloch |