Calculate effective rank of matrix

You might be interested in the following publication:

Olivier Roy and Martin Vetterli, The effective rank: A measure of effective dimensionality, 15th European Signal Processing Conference, 2007, available at https://infoscience.epfl.ch/record/110188/files/RoyV07.pdf.

They define "effective rank" as the entropy of the notional distribution obtained by normalising the singular values. It has the property that for an m x n matrix A,

1 <= erank(A) <= rank(A) <= min(m,n)

It has other pleasant properties, and a (reasonably) intuitive geometric interpretation in terms of linear transformations.