An axiomatic approach to intrinsic dimension of a dataset

Details An axiomatic approach to intrinsic dimension of a dataset An axiomatic approach to intrinsic dimension of a dataset

2007-12-12

arxiv.org/abs/0712.2063v1

Neural Networks 21, 2-3 (2008), 204-213.

Computer Science

Information Retrieval

10 pages, 5 figures, latex 2e with Elsevier macros, final submission to Neural Networks with referees' comments taken into acc

Scientific paper

We perform a deeper analysis of an axiomatic approach to the concept of intrinsic dimension of a dataset proposed by us in the IJCNN'07 paper (arXiv:cs/0703125). The main features of our approach are that a high intrinsic dimension of a dataset reflects the presence of the curse of dimensionality (in a certain mathematically precise sense), and that dimension of a discrete i.i.d. sample of a low-dimensional manifold is, with high probability, close to that of the manifold. At the same time, the intrinsic dimension of a sample is easily corrupted by moderate high-dimensional noise (of the same amplitude as the size of the manifold) and suffers from prohibitevely high computational complexity (computing it is an $NP$-complete problem). We outline a possible way to overcome these difficulties.

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