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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.

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  • $\begingroup$ This is a tricky question that probably doesn't have easy answers, depending on what exactly you mean by "approximate". You may want something like "Persistent homology": en.wikipedia.org/wiki/Persistent_homology $\endgroup$ – Nathaniel Mayer Feb 17 '19 at 22:15
  • $\begingroup$ This seems like a tricky matter to deal with, if you only consider a finite set. For instance, for any finite set of points in $\mathbb{R}^n$, we can construct a (polynomial) curve which passes through all these points... in that sense, you could consider the set to be one-dimensional. $\endgroup$ – Sambo Feb 17 '19 at 22:26
  • $\begingroup$ Thank you for the help. It seems to me that this is a question of scale actually. If I can formulate the question in the right way, I'll know what I need to answer it. Just realized that this kind of dimension assigning depends on the scale on which we are looking at our set. It may seem like dimension 0, 1 or even two if one considers the set to be annulus like. $\endgroup$ – M.Π.B Feb 19 '19 at 14:04
  • $\begingroup$ Hurewicz. and Wallman have a book, I believe considered a classic, entitled Dimension Theory. I believe it may be relevant. $\endgroup$ – Chris Custer Apr 22 '19 at 4:41

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