So, after an extensive search I found no answer for this, although it might be because I don't have the knowledge to ask the right question.
Imagine that you are given a finite set of points $S$ in $\mathbb{R}^2$ which make for an approximate unit circle. So, I know that the topological dimension of this set is zero, but I also know that it is approximately one dimensional. After a quick visualization I can infer that these points only depend approximately on the parameter $\theta$, the angle. But this is a very impractical way of understanding the problem. If I choose a set which is approximately a $S^6$ hypersphere in $\mathbb{R}^{10}$ in which I only have Euclidean coordinates of these points I cannot visualize that only six parameters (excluding noise) are of interest.
How can I compute a coordinate independent real number which gives me the approximate dimension of my set? I looked at generalizations of the matrix rank but found no computations.