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Intrinsic dimension of a dataset: what properties does one expect?

Vladimir Pestov
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Abstract

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 $Θ(n)$. 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 Chávez et al.)

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Vladimir Pestov. 2007-03-25. Intrinsic dimension of a dataset: what properties does one expect?. https://doi.org/10.1109/ijcnn.2007.4371431

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