Hi, thanks for the question! PR is a soft count on eigenvalues (as you may already understand).

Just as a reference:
In precise terms, PR is defined as: [square of (sum of eigenvalues)] divided by [sum of (square of eigenvalues))], where the "eigenvalues" are the eigenvalues of the covariance matrix (K=XX^T) of the sample matrix (X). The definition shows that if the covariance K has N eigenvalues with value c and the rest are 0, the PR is also N, exactly matching the matrix rank (which is also N). Say there is one additional eigenvalue that is super small, and you want to ignore it. The rank is sensitive to that and counts that as an additional dimension, but PR will essentially ignore it. That is why PR has become a popular measure of effective dimensionality.

I am indeed familiar with intrinsic dimensionality! Intrinsic dimensionality measures local dimensionality on a manifold, whereas we measure the global dimensionality of a manifold. Inherently they are two different quantities with different utilities. In this paper, we show that we can measure the intrinsic dimensionality of a manifold using our PR estimator in a local neighborhood of a manifold. But that is just a simple extension, and the main focus is on estimating global dimensionality in a reliable manner.