An axiomatic approach to intrinsic dimension of a dataset

Neural Networks

Volumn 21, Issue 2-3, 2008, Pages 204-213

  Pestov, Vladimir ^a  

^a UNIVERSITY OF OTTAWA   (Canada)

Author keywords

Concentration of measure; Curse of dimensionality; Features; Gromov distance; High dimensional noise; Intrinsic dimension of datasets; Random sample of a manifold; Space with metric and measure

Indexed keywords

COMPUTATIONAL COMPLEXITY; DATA STRUCTURES; MATHEMATICAL MODELS; PROBABILITY;

CONCENTRATION OF MEASURE; DATASETS; DIMENSIONALITY; GROMOV DISTANCE;

NEURAL NETWORKS;

ACCURACY; ARTICLE; ARTIFICIAL NEURAL NETWORK; GEOMETRY; MATHEMATICAL ANALYSIS; MATHEMATICAL MODEL; PRIORITY JOURNAL; PROBABILITY; STATISTICAL MODEL;

ALGORITHMS; ARTIFICIAL INTELLIGENCE; NONLINEAR DYNAMICS; PATTERN RECOGNITION, AUTOMATED;

EID: 40649090828     PISSN: 08936080     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.neunet.2007.12.030     Document Type: Article

References (23)

1. Anthony M., and Bartlett P. Neural network learning: Theoretical foundations (1999), Cambridge University Press, Cambridge
2. Beyer, K., Goldstein, J., Ramakrishnan, R., & Shaft, U. (1999). When is nearest neighbor meaningful? In Proc. 7th intern. conf. on database theory (pp. 217-235)
3. Blanchard G., Kawanabe M., Sugiyama M., Spokoiny V., and Müller K.-R. In search of non-Gaussian component of a high-dimensional distribution. Journal of Machine Learning Research 7 (2006) 247-282
4. Camastra F., and Vinciarelli A. Estimating the intrinsic dimension of data with a fractal-based method. IEEE Transactions on Pattern Analysis and Machine Intelligence 24 10 (2002) 1404-1407
5. Chávez E., Navarro G., Baeza-Yates R., and Marroquín J.L. Searching in metric spaces. ACM Computing Surveys 33 (2001) 273-321
6. Costa J., and Hero A.O. Geodesic entropic graphs for dimension and entropy estimation in manifold learning. IEEE Transactions on Signal Processing 52 8 (2004) 2210-2221
7. François D., Wertz V., and Verleysen M. The concentration of fractional distances. IEEE Transactions on Knowledge and Data Engineering 19 7 (2007) 873-886
8. Garey M.R., and Johnson D.S. Computers and intractability, a guide to the theory of NP-completeness (1979), Freeman, San Francisco
9. Gromov M. Metric structures for riemannian and non-riemannian spaces. Progress in mathematics Vol. 152 (1999), Birkhauser Verlag
10. d. In: de Raedt L., and Wrobel S. (Eds). Proc. 22nd intern. conf. on machine learning (2005), AMC Press 289-296
11. Hein, M., & Maier, M. (2007). Manifold denoising as preprocessing for finding natural representations of data. In Proc. twenty-second AAAI conference on artificial intelligence (pp. 1646-1649)
12. Kégl B. Intrinsic dimension estimation using packing numbers. Advances in neural information processing systems Vol. 15 (2003), MIT Press 681-688
13. Ledoux M. The concentration of measure phenomenon. Math. surveys and monographs Vol. 89 (2001), Amer. Math. Soc, Providence
14. Mendelson S. A few notes on statistical learning theory. In: Mendelson S., and Smola A.J. (Eds). Advanced lectures in machine learning. Lect. notes in computer sci. Vol. 2600 (2003), Springer 1-40
15. Milman V. Topics in asymptotic geometric analysis. Geometric and functional analysis special volume GAFA2000 (2000) 792-815
16. Milman V.D., and Schechtman G. Asymptotic theory of finite-dimensional normed spaces (with an Appendix by M. Gromov). Lecture Notes in Math. Vol. 1200 (1986), Springer
17. Pestov V. On the geometry of similarity search: dimensionality curse and concentration of measure. Information Processing Letters 73 (2000) 47-51
18. Pestov, V. (2007). Intrinsic dimension of a dataset: what properties does one expect? In: Proc. of the 22-nd int. joint conf. on neural networks (pp. 1775-1780)
19. Potapov A., and Ali M.K. Neural networks for estimating intrinsic dimension. Physical Review E 65(2a) 4 (2002) 046212.1-046212.7
20. Rachev S.T., and Rüschendorf L. Mass transportation problems. Volume I: Theory. Volume II: Applications (1998), Springer, NY, Berlin, Heidelberg
21. Tatti, N., Mielikainen, T., Gionis, A., & Mannila, H. (2006). What is the dimension of your binary data? In 6th international conference on data mining (pp. 603-612)
22. Traina, C. Jr., Traina, A.J.M., & Faloutsos, C. (1999). Distance exponent: A new concept for selectivity estimation in metric trees. Technical report CMU-CS-99-110. Computer Science Department, Carnegie Mellon University
23. von Luxburg U., and Bousquet O. Distance-based classification with Lipschitz functions. Journal of Machine Learning Research 5 (2004) 669-695