Computer Science – Learning
Scientific paper
2007-03-25
Proceedings of the 20th International Joint Conference on Neural Networks (IJCNN'2007), Orlando, Florida (Aug. 12--17, 2007),
Computer Science
Learning
6 pages, 6 figures, 1 table, latex with IEEE macros, final submission to Proceedings of the 22nd IJCNN (Orlando, FL, August 12
Scientific paper
We propose an axiomatic approach to the concept of an intrinsic dimension of a dataset, based on a viewpoint of geometry of high-dimensional structures. Our first axiom postulates that high values of dimension be indicative of the presence of the curse of dimensionality (in a certain precise mathematical sense). The second axiom requires the dimension to depend smoothly on a distance between datasets (so that the dimension of a dataset and that of an approximating principal manifold would be close to each other). The third axiom is a normalization condition: the dimension of the Euclidean $n$-sphere $\s^n$ is $\Theta(n)$. We give an example of a dimension function satisfying our axioms, even though it is in general computationally unfeasible, and discuss a computationally cheap function satisfying most but not all of our axioms (the ``intrinsic dimensionality'' of Ch\'avez et al.)
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