Intrinsic dimension of a dataset: what properties does one expect?
<|reference_start|>Intrinsic dimension of a dataset: what properties does one expect?: 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.)<|reference_end|>
@article{pestov2007intrinsic,
    title={Intrinsic dimension of a dataset: what properties does one expect?},
    author={Vladimir Pestov},
    journal={Proceedings of the 20th International Joint Conference on Neural
  Networks (IJCNN'2007), Orlando, Florida (Aug. 12--17, 2007), pp. 1775--1780.},
    year={2007},
    doi={10.1109/IJCNN.2007.4371431},
    archivePrefix={arXiv},
    eprint={cs/0703125},
    primaryClass={cs.LG}
}