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.)