A colleague in applied statistics sent me this:
"I was wondering if you know any way to find out the true dimension of the domain of a function. For example, a circle is a one dimensional function in a two dimensional space. If I do not know how to draw, is there a statistic that I can compute that tells me that it is a one dimensional object in a two dimensional space? I have to do this in high dimensional situations so cannot draw pictures. Any help will be greatly appreciated."
The notion of dimension here is obviously ill-defined. I mean, I could run a curve through any finite collection of points in high dimensional space, and say that my data is one-dimensional. But, depending on the configuration, there may be an easier or more efficient way to describe the data as a higher dimensional set.
Such issues must have been considered in the statistics literature, but I'm not familiar with it. Any pointers or suggestions? Thanks!