# Geometric Structure of High-Dimensional Data

• Jianzhong Wang

## Abstract

In applications, a high-dimensional data is given as a discrete set in a Euclidean space. If the points of data are well sampled on a manifold, then the data geometry is inherited from the manifold. Since the underlying manifold is hidden, it is hard to know its geometry by the classical manifold calculus. The data graph is a useful tool to reveal the data geometry. To construct a data graph, we first find the neighborhood system on the data, which is determined by the similarity (or dissimilarity) among the data points. The similarity information of data usually is driven by the application in which the data are used. In this chapter, we introduce the methods for defining the data similarity (or dissimilarity). We also introduce the preliminary spectral graph theory to analyze the data geometry. In Section 1, the construction of neighborhood system on data is discussed. The neighborhood system on a data set defines a data graph, which can be considered as a discrete form of a manifold. In Section 2, we introduce the basic concepts of graphs. In Section 3, the spectral graph analysis is introduced as a tool for analyzing the data geometry. Particularly, the Laplacian on a graph is briefly discussed in this section. Most of the materials in Sections 2 and 3 are found in [1–3].

## Keywords

Fast Fourier Transform Geometric Structure Adjacency Matrix Undirected Graph Weighted Graph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
Bondy, J., Murty, U.: Graph Theory. Springer (2008).Google Scholar
2. [2]
Chartrand, G.: Introductory Graph Theory. Dover (1985).Google Scholar
3. [3]
Chung, F.R.: Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, No. 9. AMS (1996).Google Scholar
4. [4]
Shakhnarovich, G., Darrell, T., Indyk, P. (eds.): Nearest-Neighbor Methods in Learning and Vision, Theory and Practice. MIT (2006).Google Scholar
5. [5]
Bachmann, C.M., Ainsworth, T.L., Fusina, R.A.: Improved manifold coordinate representations of large-scale hyperspectral scenes. IEEE Trans. Geo. Remote Sensing 44, 2786–2803 (2006).
6. [6]
Bozkaya, T., Ozsoyoghu, M.: Distance-based indexing for highdimensional metric spaces. In: Proc. ACM SIGMOD, p. 357–368 (1997).Google Scholar
7. [7]
Friedman, J.H., Bentley, J.L., Finkel, R.A.: An algorithm for finding best matches in logarithmic expected time. ACM Trans. Math. Softw. 3(3), 209–226 (1977).
8. [8]
Katayama, N., Satoh, S.: The SR-tree: An index structure for high-dimensional nearest neighbor queries. Proc. ACM SIGMOD p. 369–380 (1997).Google Scholar
9. [9]
Kim, B.S., Park, S.B.: Fast nearest neighbor finding algorithm based on ordered partition. IEEE Trans. Pattern Anal. Mach. Intell. 8(6), 761–766 (1986).
10. [10]
Lubiarz, S., Lockwood, P.: Evaluation of fast algorithms for finding the nearest neighbor. Proc. IEEE Int. Conf. Acoust., Speechand Signal Process. 2, 1491–1494 (1997).Google Scholar
11. [11]
McNames, J.: A fast nearest-neighbor alogorithm based on a principal axis search tree. IEEE Trans. Pattern Anal. Mach. Intell. 23(9), 964–976 (2001).
12. [12]
Yianilos, P.N.: Data structure and algorithms for nearest neighbor search in general metric spaces. Proc. ACM-SIAMSymp. Discr. Algorithms p. 311–321 (1993).Google Scholar
13. [13]
Chui, C.K.: An Introduction to Wavelets, Wavelet Analysis and its Applications, vol. 1. Academic Press, Inc. (1992).Google Scholar
14. [14]
Chui, C.K., Wang, J.Z.: A cardinal spline approach to wavelets. Proc. Amer. Math. Soc. 113, 785–793 (1991).
15. [15]
Chui, C.K., Wang, J.Z.: On compactly supported spline wavelets and a duality principle. Trans. Amer. Math. Soc. 330, 903–915 (1992).
16. [16]
Chui, C.K.: Wavelets: A Mathematical Tool for Signal Analysis. SIAMMonographs on Mathematical Modeling and Computation. Society for Industrial and Applied Mathematics, Philadelphia (1997).Google Scholar
17. [17]
Chui, C.K., Wang, J.Z.: A general framework of compactly supported splines and wavelets. J. Approx. Theory 71(3), 263–304 (1992).
18. [18]
Laub, J., Müller, K.R.: Feature discovery in non-metric pairwise data. Journal of Machine Learning Research 5, 801–818 (2004).
19. [19]
Mahalanobis, P.C.: On the generalised distance in statistics. Proceedings of the National Institute of Sciences of India 2(1), 49–55 (1936).
20. [20]
De Maesschalck, R., Jouan-Rimbaud, D., Massart, D.: The mahalanobis distance. Chemometrics and Intelligent Laboratory Systems 50, 1–8 (2000).